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Full text of "Nonlinear static and dynamic finite element analysis of multilayer shell structures"

NONLINEAR STATIC AND DYNAMIC FINITE ELEMENT ANALYSIS OF 
MULTILAYER SHELL STRUCTURES 




BY 

XIANGGUANG TAN 



A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL 
OF UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT 
OF THE REQUIREMENTS FOR THE DEGREE OF 
DOCTOR OF PHILOSOPHY 

UNIVERSITY OF FLORIDA 



2002 



To my parents. 



ACKNOWLEDGMENTS 

I wish to express my sincere gratitude to my advisor, Prof. Loc Vu-Quoc, for his pa- 
tience, guidance, support and friendship throughout my Ph.D. education at the University 
of Florida. I have greatly benefited from his stimulating approach to research and his re- 
lentless pursuit of perfection in organization and documentation. Many thanks are extended 
to him for his invaluable help in preparing this LINUX/LaTeX document. 

I also wish to acknowledge the members of my examining committees, Professors 
Martin A. Eisenberg, Raphael T. Haftka, Marc Hoit, Andrew J. Kurdila, and W. Gregory 
Sawyer for their careful examination of the dissertation, and their invaluable comments 
and insights, which made a deep impact on my research. I also benefited greatly from their 
graduate courses and from their help in many other aspects. 

I am indebted to several colleagues and mentors for their help in my present work: 
in particular, Hui Deng for the use of finite element code, FEAP and many insightful dis- 
cussions on the geometrically-exact shell theory; Fuller L. Brian for the installation of the 
LLNL package; Paul Dionne, Andrzej Przekwas, Marek Turowski, and H.Q. Yang at the 
CFDRC for discussion of the model reduction technique; Prof. Chen-Chi Hsu to work for 
him as his teaching assistant; and to my friends, Joakim Andersson, Jonas Bjornstrom, Mat- 
tias Horling, Stefan Jansson, Kil-Soo Mok, Mattias Quas, Simon. Sjogren, Xiang Zhang, 
and Yuhu Zhai, and many others, who have made my stay at Gainesville one of the most 
memorable periods of my life. 

Last, but certainly not least, my heartfelt thanks go to my parents for their love, and 
encouragement through my life. I am indebted to my girlfriend, Veronica Leung. Without 
her love and care, I could not have accomplished so much. 

This research is supported by a grant from the National Science Foundation, and also 
by the CFDRC. This support is gratefully appreciated. 

iii 



TABLE OF CONTENTS 

page 

ACKNOWLEDGMENTS iii 

ABSTRACT viii 

CHAPTER 

1 OVERVIEW 1 

1 . 1 Objectives and Motivation 1 

1.1.1 Formulation and Kinematics 2 

1.1.2 Computational Aspects 4 

1.2 Chapter Overview 5 

2 GEOMETRICALLY-EXACT SANDWICH SHELL FORMULATION 7 

2.1 Introduction 7 

2.2 Virtual Powers 8 

2.2.1 Basic Kinematic Assumptions and Configurations 9 

2.2.2 Virtual Powers 11 

2.2.2.1 Power of contact forces/couples and conjugate strain mea- 
sures 11 

2.2.2.2 Power of assigned forces/couples 13 

2.2.3 Constitutive Relations 13 

2.3 Weak Form and Linearization 14 

2.3.1 Admissible Variations, Tangent Spaces 14 

2.3.2 Weak Form of Equations of Equilibrium 15 

2.3.3 Contact Weak Form 16 

2.3.4 Assigned Weak Form 16 

2.3.5 Linearization of Contact Weak Form 17 

2.3.5.1 Update of inextensible directors 17 

2.3.5.2 Perturbed configuration 18 

2.3.5.3 Linearized strain measures 20 

2.3.5.4 Linearized contact weak form 22 

2.3.6 Matrix-Operator Format of Contact Weak Form 22 

2.3.6.1 Material tangent operator 31 

2.3.6.2 Geometric tangent operator 34 

2.4 Numerical Examples for Statics of Sandwich Shells 46 

2.4.1 Roll-down Maneuver of a Sandwich Plate 48 

iv 



2.4.2 Sandwich Plate with Ply Drop-offs 49 

2.4.2.1 Sandwich plate with ply drop-off 49 

2.4.2.2 Two-layer plate with ply drop-off: aspect ratio A = 5 : 1 : 
(1,0.5) 53 

2.4.2.3 Two-layer plate with ply drop-off: aspect ratio A = 20 : 1 : 
(1,0.5) 53 

2.4.2.4 Two-layer plate with ply drop-off: aspect ratio A = 20 : 10 : 
(1,0.5) 61 

3 OPTIMAL SOLID SHELLS FOR NONLINEAR ANALYSES OF MULTILAYER 

COMPOSITES : STATICS 68 

3.1 Introduction 68 

3.2 Kinematic Assumption and FHW Variational Formulation 74 

3.2.1 Kinematics of Solid-Shell in Curvilinear Coordinates 74 

3.2.2 Variational Formulation of EAS Method 78 

3.3 Finite-Element Discretization 83 

3.3.1 The Weak Form of Modified Two-Field FHW Functional 83 

3.3.2 Spatial Discretization 84 

3.3.3 Linearization of the Discrete Weak Form . . . 85 

3.3.4 Material Law in Convected Basis 89 

3.3.5 The ANS Method 93 

3.3.5.1 Transverse shear strains 93 

3.3.5.2 Transverse normal strain 94 

3.4 Interpolation of the Enhanced Strains 94 

3.4.1 The Regular Enhanced Strains Treatment 95 

3.4.2 Proposed Efficient Enhancing Strains 100 

3.4.3 Equivalence Between EAS Element and Incompatible Mode Element 102 

3.4.3.1 Tensor form of enhancing strains 103 

3.4.3.2 Equivalence of condensed stiffness matrices 107 

3.5 Numerical Examples 109 

3.5. 1 Patch Tests and Optimal Number of Parameters 110 

3.5.1.1 Membrane patch test Ill 

3.5.1.2 Out-of-plane bending patch test Ill 

3.5.2 Cantilever Plate 113 

3.5.2.1 Cantilever beam: in-plane bending 114 

3.5.2.2 Cantilever plate: out-of-plane bending 115 

3.5.3 In-plane Bending Problem with Nearly Incompressibility 119 

3.5.4 Snap-through of a Shallow, Cylindrical Roof under a Point Load . . 121 

3.5.5 Pinched Hemispherical Shell 122 

3.5.6 Multilayer Composite Plate 125 

3.5.6.1 Two-layer composite plate: linear solution 125 

3.5.6.2 Multilayer composite plate with ply drop-offs 126 

3.5.7 Multilayer Composite Hyperbolical Shell with Ply Drop-offs .... 129 



v 



4 OPTIMAL SOLID SHELLS FOR NONLINEAR ANALYSES OF MULTILAYER 



COMPOSITES : DYNAMICS 132 

4.1 Introduction 132 

4.2 Dynamics of Solid Shells by an EM Conserving Algorithm 133 

4.2.1 Time Discretization on Dynamic Weak Form 134 

4.2.2 Linearization of Dynamic Weak Form 136 

4.3 Enhanced-Assumed-Strain Method Based on Deformation Gradient . . . 143 

4.3.1 Weak Form 143 

4.3.2 Finite Element Discretization and Linearization 147 

4.3.3 Assumed Natural Strain (ANS) Treatment 150 

4.3.4 Simplified Formulation 151 

4.4 Numerical Examples 153 

4.4. 1 Double Cantilever Elastic Beam under Point Load 154 

4.4.2 Pinched Cylindrical Multilayer Shell 157 

4.4.3 Free-Flying Single-Layer Plate 159 

4.4.4 Free-Flying Multilayer Plate with Ply Drop-offs 161 

5 EFFICIENT AND ACCURATE MULTILAYER SOLID-SHELL ELEMENT: NON- 
LINEAR MATERIALS AT FINITE STRAIN 175 

5.1 Introduction 175 

5.2 Nonlinear Material Law 180 

5.2.1 The Mooney-Rivlin Material Models 180 

5.2.2 The Hyperelastoplastic Model 182 

5.2.2.1 Multiplicative decomposition of the deformation gradient F 183 

5.2.2.2 Spectral form based on the right Cauchy-Green tensor C . 185 

5.3 Explicit Time Integration Method for Solid-Shell Elements 193 

5.4 Numerical Examples 196 

5.4.1 Large Deformation of Rubber Shells 197 

5.4. 1 . 1 Stretch of a rubber sheet with a hole 1 98 

5.4.1.2 The snap-through of a conic shell 198 

5.4.1.3 Large motion of the pinched cylindrical shell 200 

5.4.1.4 Rubber hemispherical shell 203 

5.4.2 Large Deformation of Elastoplastic Shells 204 

5.4.2.1 Bending of a cantilever beam 206 

5.4.2.2 Elastoplastic response of a channel beam 208 

5.4.2.3 Pinched hemisphere 211 

5.4.2.4 Elastoplastic response of a simply supported plate 213 

5.4.2.5 Elastoplastic response of a pinched cylinder 215 

5.4.2.6 Free-flying multilayer plate with ply drop-offs 218 

5.4.2.7 The impact of a boxbeam 221 

5.4.2.8 Pipe whip 223 



6 SOLID SHELL ELEMENT FOR ACTIVE PIEZOELECTRIC SHELL STRUC- 

vi 



TURES AND ITS APPLICATIONS 228 

6.1 Introduction 228 

6.2 The Solid-Shell Formulation 231 

6.2.1 The Kinematics of Piezoelectric Solid-Shell Formulation 231 

6.2.2 Piezoelectric Solid-Shell Element 234 

6.2.2.1 Functional and finite element formulation 234 

6.2.2.2 Linear piezoelectric material law in convected coordinate . 239 

6.2.3 Composite Solid-Shell Element 242 

6.3 Simulation Control Design 244 

6.3.1 Finite Element System Equation of Piezoelectric Structure 244 

6.3.2 Reduced-Order Model of Piezoelectric Finite Element System . . . 246 

6.3.3 Controller Design 249 

6.4 Numerical Examples 252 

6.4.1 Cantilever Plate: Out-of-Plane Bending 253 

6.4.2 Multilayer Composite Hyperbolical Shell . 255 

6.4.3 Piezoelectric Bimorph Beam 256 

6.4.4 Cantilever Plate with PZT Actuators 259 

6.4.5 Cantilever Plate with PZT Actuator and Sensor 263 

7 CLOSURE 268 

7.1 Conclusion 268 

7.2 Directions for Future Research 270 

APPENDED 

A SOLID-SHELL FORMULATION 272 

A. 1 Finite Element Approximation of Solid-Shell Element 272 

A.2 Solution Procedure of Nonlinear Equations 279 

A.3 Explicit Integration Algorithm with EAS Method 280 

A.4 Return Mapping Algorithm for J 2 Flow Theory with Isotropic Hardening . 280 

A.5 Elastoplastic Moduli S-J 281 

A. 6 Algorithmic Moduli for Return Mapping 282 

B PIEZOELECTRIC SOLID-SHELL FORMULATION 285 

B. l Model Reduction Algorithm 285 

B.2 Solving Procedure on Control Design 285 

REFERENCES 287 

BIOGRAPHICAL SKETCH 302 

vii 



Abstract of Dissertation Presented to the Graduate School 
of the University of Florida in Partial Fulfillment of the 
Requirements for the Degree of Doctor of Philosophy 

NONLINEAR STATIC AND DYNAMIC FINITE ELEMENT ANALYSIS OF 
MULTILAYER SHELL STRUCTURES 

By 

Xiangguang Tan 
August, 2002 

Chairman: Loc Vu-Quoc 

Major Department: Aerospace Engineering, Mechanics, and Engineering Science 

Firstly, the geometrically-exact sandwich shell formulation is developed to analyze 
sandwich shells undergoing large deformation. Finite rotation of the director in each layer 
is allowed, with shear deformation independently accounted for in each layer. The thick- 
ness and the length of each layer can be arbitrary, thus allowing the modeling of multilayer 
structures having ply drop-offs. The weak form of governing equations is constructed, and 
the linearization and inextensible directors update are derived. Numerical examples on 
elastic sandwich plates are presented to illustrate salient features of the formulation. 

Furthermore, we present a low-order solid-shell element formulation — having only 
displacement degrees of freedom (dofs) (i.e., without rotational dofs)— that has an opti- 
mal number of parameters to pass the plate patch tests (both membrane and out-of-plane 
bending), thus allowing for efficient and accurate analyses of large deformable multilayer 
shell structures. The formulation is based on the mixed Fraeijs de Veubeke-Hu-Washizu 
(FHW) variational principle leading to a novel enhancing assumed strain (EAS) tensor, 
with improved in-plane and out-of-plane bending behaviors (Poisson thickness locking). 
Shear locking and curvature thickness locking are treated using the Assumed Natural Strain 
(ANS) method. We provide an optimal combination of the ANS method and the minimal 
number of EAS parameters to pass the out-of-plane bending patch test and treat the locking 



viii 



associated with (nearly) incompressible materials. The energy-momentum (EM) conserv- 
ing algorithm for the current element is presented. Two nonlinear 3-D material models are 
applied directly without requiring the enforcement of the plane-stress assumption. More- 
over, we present a low-order accurate piezoelectric solid-shell element formulation for 
piezoelectric sensors and actuators used in active shell structures. Numerical examples in- 
volving static analyses and implicit/explicit dynamic analyses of multilayer shell structures 
having a large range of element aspect ratios for both material and geometric nonlinearities 
are presented. Numerical examples involving static analyses and active vibration control of 
piezoelectric shell structures are also presented. The developed element formulations are 
accurate and efficient in modeling and analyzing general nonlinear multilayer composite 
shell structures. 




CHAPTER 1 
OVERVIEW 

Shells and shell structures are thin-walled, generally curved bodies in a three-dimens- 
ional space. Their load-bearing behavior is dominated by stretching and bending. Shell 
structures with different layers in the thickness direction are generally addressed as multi- 
layer shells. For a comprehensive and valuable history and review of linear and nonlinear 
shell theories, see Timoshenko and Woinowsky-Krieger [1959], Naghdi [1972] and Basar 
and Kratzig [2000]. Below we describe the objectives and motivation for the current re- 
search on multilayer shells. Some of the motivating factors behind the present work and 
literature review are delineated in the following chapters. 

1.1. Objectives and Motivation 

Multilayer shell structures have widespread applications in engineering. Laminated 
composite structures, initially developed for use in the aerospace industry, have played 
an increasingly important role in robotics and machine systems that require high operat- 
ing speed. The low weight and high stiffness offered by laminated composite structures 
help reduce power consumption, increase the ratio of payload/self-weight, and improve 
the accuracy of motion characteristics and reduce the level of acoustic emission of these 
systems. It is shown from computer simulations with experimental corroboration that the 
low weight/stiffness ratio of laminated composites is essential for obtaining high perfor- 
mance in slider-crank and four-bar linkage systems (Sung, Thompson, Crowley and Cuccio 
[1986], Thompson and Sung [1986]). More recently, considerable attention has been given 
to a class of active structures with embedded piezoelectric layers as sensors and actuators 
(Evseichik [1989], Tzou [1989], Saravanos, Heyliger and Hopkins [1997]) or interfero- 
metric optical fiber sensors (Sirkis [1993]) for monitoring the strain level and for vibration 
control. Large overall motion of multilayer structures can be found in robot arms or space 



1 



2 



structures with embedded sensors/actuators. Another example of multilayer structures can 
be found in the damping of structural vibration by using viscoelastic constrained layers (Al- 
berts [1993], Dubbelday [1993], Rao [1993]) (Figure 1.1). The use of sandwich plates to 
absorb energy in crashes (car, train, airplane) was investigated by Goldsmith and Sackman 
[1991]. 




Figure 1.1. Multilayer shells with patches of constrained viscoelastic materials or of piezo- 
electric materials. 

The design and analysis of multilayer shell structure is a major challenge that in- 
volves the proper modeling of composite materials with highly anisotropic properties, com- 
plex geometric configuration, and strongly nonlinear material behavior. For example, only 
a few studies so far have been performed on large deformation analysis of 3-D nonlin- 
ear composite laminates. There have been no analytical studies involving 3-D analysis of 
multilayer shells with nonlinear material behavior and large deformation. 
1.1.1. Formulation and Kinematics 

For nonlinear analysis of multilayer shell structures, we developed two different finite 
element formulations: the geometrically-exact 1 multilayer shell formulation and multilayer 
solid-shell formulation. 

In the geometrically-exact multilayer shell formulation, the 3-D analysis is reduced 

to a set of 2-D stress-resultant equations based on the kinematic assumptions. This model 

accommodates large deformation and large overall motion. The layer directors at a point in 

1 The term "geometrically-exact" reflects the fact that no additional kinematic assumptions are made 
beyond the one-director assumption. In particular, approximations of the type sinO es 9 - 3 /6 are entirely 
avoided. 



3 



the reference surface are connected to each other by universal joints, as in a chain of rigid 
links. The thickness and length of each layer can be arbitrary, thus making it suitable to 
model shell structures with ply drop-offs. The equations of motion of the multilayer shell 
are derived based on the principle of virtual power, and expressed in terms of weighted 
resultant forces and couples. The overall deformation of a sandwich shell can be described 
by the deformation of a reference layer (which can be any layer; not necessarily the middle 
layer). The unknown kinematic quantities are therefore the three displacement components 
of the centroidal surface of the reference layer and two rotational components for each 
layer director. No restriction is imposed on the magnitude of the displacement field, whose 
continuity across the layer interfaces is exactly enforced. Finite rotations of the directors in 
each layer are allowed, with shear deformation independently accounted for in each layer. 
We have implemented the geometrically-exact sandwich shell element to illustrate the ver- 
satility of formulation in the large deformable multilayer shell analysis involving linear 
elastic material and small strain. Due to the kinematic assumptions, the present formula- 
tion is more accurate than the equivalent single-layer shell models in the interlaminar stress 
analysis, especially for thick and moderate thick shells. 

In the solid-shell formulation, on the other hand, the shell kinematic descriptions 
used are the displacement of the top and bottom surface of the shell. All kinematic quan- 
tities such as displacements and the corresponding strains can be finite. For multilayer 
shells, one solid-shell element in the thickness direction can be used for either one material 
layer or several layers. In contrast to the shell formulation based on the degenerated shell 
concept and the classical shell theory, the present element can incorporate the complex 3-D 
material models without enforcing the zero transverse normal stress condition, can avoid 
complex update algorithms for finite rotations, and can account for the transverse normal 
stress. Based on the mixed Fraeijs de Veubeke-Hu-Washizu (FHW) variational principle, 
the present low-order solid-shell element is designed to pass the plate patch tests and to 
remedy volumetric locking, therefore allowing efficient and accurate nonlinear analyses 
of multilayer shell structures. Moreover, the kinematic description provides a natural way 



4 



to connect solid-shell elements to regular solid elements without the need for transition 
elements; such feature can also benefit the detailed modeling of shells with patches of 
piezoelectric or viscoelastic materials. For the interlaminar stress analysis, with the refine- 
ment through the thickness, the solid-shell element model can determine the localized 3-D 
stress field (e.g., delamination, free-edge effect) accurately. 
1 . 1 .2. Computational Aspects 

Several aspects can directly contribute to the success and generality of numerical 
simulations: 1) element formulations; 2) time-integration schemes; and 3) equation solu- 
tion strategies. The geometrically-exact sandwich shell formulation uses the resultant form 
to avoid numerical integration in the thickness direction for elastic materials. The solid- 
shell formulation uses the numerical integration for general nonlinear constitutive models. 
All kinematic quantities such as displacements can be finite, and the update procedure is 
proceeded in an exact manner, without approximations. 

Engineering applications mandate the use of relatively coarse meshes for complex 
geometries. The development of convergent elements, which are free of spurious numer- 
ical locking, are variationally consistent, achieve good accuracy with coarse meshes, and 
satisfy stability and completeness requirements, is essential. Flexural super-convergence 
in membrane deformation is also important for applications involving in-plane bending. 
Moreover, the use of low-order interpolations is extremely desirable for their simplicity, 
efficiency and amenability to contact implementations. To this end, we use the methods 
of enhanced assumed strains (EAS) and assumed natural strains (ANS) judiciously to con- 
struct low-order elements possessing the above features for the analysis of multilayer shells. 

In this work, we have implemented a number of dynamic time-stepping implicit/explicit 
algorithms in the context of the present formulations for transient integration of the result- 
ing semi-discrete finite element equations. The time step-size for the implicit integration 
can be much larger than that for the explicit integration. The explicit method, on the other 
hand, needs much less computational effort at each time step since the matrix factoriza- 
tion is not needed. For elastodynamics, the introducing of numerical damping is essential 



5 



to increase the numerical stability of implicit integration methods, even for the energy- 
momentum conserving algorithm. 

The solution of discrete equations for problems involving large deformation and 
long-term simulations can be accomplished with the Newton-Raphson scheme. The numer- 
ical efficiency of this approach is a byproduct of the asymptotically quadratic convergence 
of its iterations. To maintain this rate of convergence, the exact linearization of discrete 
equations is explicitly obtained and implemented in the present work. For nonlinear mate- 
rials, the consistent tangent moduli are crucial to be derived. For the quasi-static analysis of 
unstable systems, arc-length method is used to find stability points and trace post-buckling 
paths. Based on the above solution strategies, a large time or load increment is allowed to 
use, while a good balance of accuracy and efficiency is maintained. 

1.2. Chapter Overview 

This dissertation is divided into six chapters. Two finite element models for multi- 
layer shell structures, the geometrically-exact sandwich shell element and the solid-shell 
element, are formulated and implemented. 

Chapter 2 presents the static analysis of the geometrically-exact sandwich shell el- 
ement formulation. The kinematic description and equilibrium equations of the sandwich 
shell model are presented in Section 2.2. The corresponding weak form and linearization 
are given in Section 2.3. Numerical examples for statics of sandwich shells are shown in 
Section 2.4. This chapter has been published by Vu-Quoc, Deng and Tan [2000]. Readers 
refer to Vu-Quoc, Deng and Tan [2001] for the corresponding dynamic analysis. 

In Chapter 3, we carry out the static analysis of the optimal solid-shell element for- 
mulation for multilayer composites. After a presentation of the kinematics assumption 
and the formulation based on the Fraeijs de Veubeke-Hu-Washizu (FHW) variational prin- 
ciple (Felippa [2000]) in Section 3.2, we discuss the finite-element discretization and its 
implementation in Section 3.3. A review of the EAS method together with our proposed 
modification is presented in Section 3.4. We present the numerical results in Section 3.5. 
This chapter will be published by Vu-Quoc and Tan [2002a]. 



6 



Chapter 4 addresses the dynamic analysis of the optimal solid-shell element formu- 
lation for multilayer composites. We devoted Section 4.2 to the dynamic aspect and the use 
of the energy-momentum algorithm for elastic materials. A variant of the EAS formulation 
based on the deformation gradient (instead of Green-Lagrange strains) for solid shells is 
the focus of Section 4.3. Numerical results are shown in Section 4.4. This chapter will be 
published by Vu-Quoc and Tan [20026] . 

In Chapter 5, we present static and dynamic analyses of the multilayer solid-shell 
element formulation for nonlinear materials at finite strain. Two nonlinear material models 
(i.e., Mooney-Rivlin material and hyperelastoplastic material), and their implementations 
are discussed in Section 5.2. The explicit integration method for solid-shell elements is 
addressed in Section 5.3. Numerical simulations, which illustrate the performance of the 
proposed element formulation, and exhibit both material and geometric nonlinearities in the 
large-scale implicit/explicit analyses, are given in Section 5.4. This chapter was submitted 
for the publication by Tan and Vu-Quoc [2002a]. 

Chapter 6 discusses the solid shell element for active piezoelectric shell structures 
and its applications. In Section 6.2, we introduce the kinematics and variational formu- 
lation of the piezoelectric solid-shell element, and then present the composite solid-shell 
element. The control design for structures with piezoelectric sensors and actuators is dis- 
cussed in Section 6.3. Numerical simulations that illustrate the performance of the pro- 
posed formulations, including comparisons with available experiment results and solutions 
obtained from shell elements and solid elements, are given in Section 6.4. This chapter was 
submitted for the publication by Tan and Vu-Quoc [20026]. 

Chapter 7 gives the closure of our work. Conclusions are drawn in Section 7.1 and 
directions for future investigation are suggested in Section 7.2. 



CHAPTER 2 

GEOMETRICALLY-EXACT SANDWICH SHELL FORMULATION 

2.1. Introduction 

Sandwich structures have played an important role in several areas of engineering. 
Many background references were cited by Vu-Quoc and Ebcioglu [1995], and are not re- 
peated in the present follow-up work, except for particularly relevant ones. We refer to 
review papers such as Reddy [1989], Noor and Burton [1989], Noor [1990], Reddy and 
Robbins [1994], and the references therein for various aspects on formulations for multi- 
layer structures. The accuracy of layerwise theory, as compared to single-layer theory with 
a shear correction factor, was demonstrated amply by Reddy [1989], where a comparison 
of transverse shear stress with 3-D elasticity solution was provided (Reddy [1993]). We de- 
scribe here a continuation of the results reported by Vu-Quoc, Ebcioglu and Deng [1997], 
where the equations of motion for geometrically-exact sandwich shells are derived. Focus- 
ing on the static case in the present work, we develop a Galerkin projection of the resulting 
nonlinear governing equations of equilibrium. 

In the present formulation, each layer in a sandwich shell structure can have different 
thickness and side lengths. As such, the present formulation can be used to model an 
important class of multilayer structures with ply drop-offs. Another important application 
of the present formulation is the modeling of shell structures with patches of constrained 
viscoelastic materials and/or patches of piezoelectric materials. No restriction is imposed 
on the magnitude of the displacement field, whose continuity across the layer interfaces 
is exactly enforced. Finite rotations of the directors in each layer are allowed, with shear 
deformation independently accounted for in each layer. The layer directors at a point in the 
reference surface are connected to each other by universal joints, and form a chain of rigid 
links. The overall deformation of a sandwich shell can be described by the deformation of 



7 



8 



a reference layer. The unknown kinematic quantities are therefore the three displacement 
components of the deformed reference surface, and the unit directors associated with the 
layers. 

The starting point for the development of the Galerkin projection of the governing 
equations of equilibrium is a nonlinear weak form based on the stress power of a sandwich 
shell, from which the expressions of fully nonlinear strain measures are obtained (Vu- 
Quoc et al. [1997]). A linearization of this nonlinear weak form is performed for use in the 
solution for the kinematic quantities via the Newton-Raphson method. Together with the 
update of the inextensible directors, the linearization leads to a symmetric tangent stiffness 
operator, which is composed of a geometric part and a material part. The consistency 
in the linearization leads to a quadratic rate of asymptotic convergence in the Newton- 
Raphson iterative solution. Linear finite element functions are chosen to form a basis for 
the Galerkin projection of the linearized equilibrium equations into a finite-dimensional 
subspace of trial solutions. The tangent stiffness matrix is symmetric, and is evaluated 
using selectively reduced integration in all layers to avoid shear locking. 

Several numerical examples, including the bending and torsion of a sandwich plate, 
are presented to illustrate the salient features of the present formulation. In particular, the 
important case of sandwich shells with ply drop-offs under large deformation is presented. 
Results are compared with those obtained using the commercial nonlinear finite element 
code ABAQUS [1995]. 

2.2. Virtual Powers 

In this section, we summarize the kinematic description of the sandwich shell model 

developed in Vu-Quoc et al. [1997], and the equilibrium equations in weighted resultant 

form. 2 The component form of the stress power and of the constitutive relation are also 

given at the end of this section. 

2 The word "weighted" here is used to indicate that the resultants are in "weighted" tensor form and not 
in true tensor form (the reader is referred to Vu-Quoc et al. [1997] for an explanation). 



9 



2.2.1. Basic Kinematic Assumptions and Configurations 

1 

Let A designate the material surface of the shell, and H := (J t£\H the total 
thickness of the sandwich shell (Figure 2.1), where (qH is the thickness of layer (£). Let 
<&o : A x H i— > B Q be the mapping from the material configuration to the initial (reference) 
configuration, where A x H = B is the material configuration of the sandwich shell. The 
material domain for layer (£) is denoted by ^B such that 

l 

W B :=Ax [t) H , B:= |J (e) B . (2.1) 

t=-\ 




2 



Figure 2.1. Sandwich shell: Profile and geometric quantities. 

Let $ : A x H i-> B t be the deformation map from the material configuration to 
the current deformed (spatial) configuration. We use the notation £ := { £ 3 } e B to 
denote the coordinates of a material point, where £- := { f 1 , ^ 2 } 6 .4 is referred to as the 
material surface coordinates, and £ 3 e H the material through-the-thickness coordinate. 
The deformation map for each of the three layers is written as follows 

»*(*,*):= (e)V(e,t) + (e - {t) Z) m t , for ^= -1,0,1, (2.2) 



10 



where ^<p : A h* R 3 is the deformation map of the centroidal surface of layer (£), and 
(£)t : A ' — * 5 2 the unit director (represent transverse fiber vector) associated with layer 
(£), (f)Z the distance from the centroidal surface of layer (£) to the centroidal surface of 
the reference layer (0) (for which (o)Z = 0). The centroidal surface of each layer (£), 
which does not necessarily correspond to the geometric center of the cross-section of layer 
(£) are defined as follows: Let ^)p and ^)p t be the mass density in the initial and the 
spatial configuration, B and B t , respectively. We select the centroidal surface (e)<P of 
layer (£) such that 

/ (e + (-dz) i-Dio ( - l)Po de = J (e + (-dz) (-^jti-DPtde =o, 

/ £ 3 (o)j (o)Pod^ 3 = I £ 3 ( o)Jt wPtd? =0, (2.3) 
(o) 7 * (0)W 

/ (^ 3 - (i)^) (i)i (i)P ^^ 3 = / (e 3 - m Z) (Dftd^ 3 =0. 

(1)W 

Using the assumption that the layer directors behave like a chain of rigid links connected to 
each other by universal joints, the deformation maps of the centroidal surface of the outer 
layers (1) and (-1) can be related to the deformation map of the centroidal surface of the 
reference layer (0) as 

(i)V (€ a , t) := (0 )V> t) + {0 )h + ( o)t + , 

(-i)V := ( )<p t) - (0) h~ (0) t - ( _i)t , (2.4) 

where and (^/i - are the distances from the centroidal surface of layer (£) to its top 
surface and to its bottom surface, respectively. The deformation of the sandwich shell is 
therefore described by four vector-valued mappings collectively denoted by 

* := {(0)<P, (e)t, for £= -1,0, l} . (2.5) 

In terms of components, we have three components for (0 )V?, and two components for 
each inextensible director w t, thus leading to a total of nine components, which are the 
principal kinematic unknowns to be solved for. 



1 1 



The deformation gradient for layer (I) is 



{t)F := 



GRAD w $ 



GRAD (£)$ . (2.6) 



The Jacobian determinants of the mapping (/)4> , and are given below 

{e) j := det[GRAD (0 * o (£)] , (2.7) 
w j ( := det[GRAD w * (£,*)]. (2.8) 

2.2.2. Virtual Powers 

Here we summarize the expressions for the power 7 C of contact forces/couples and 
for the power 7 a of assigned forces/couples. Together these powers play a crucial role in 
the derivation of the equations of motion (see Vu-Quoc et al. [1997] for more details). The 
balance of the power of contact forces/couples and the power of assigned forces/couples 
as expressed by T c = CP a leads to the equation of equilibrium for geometrically-exact 
sandwich shells (see Eq. (84) and Eq. (85) of Vu-Quoc et al. [1997]). 
2.2.2.1. Power of contact forces/couples and conjugate strain measures 

The set of convected basis vectors on the spatial (current) centroidal surface are de- 
noted by | (f)<Zj j, where the underlined index i is to be expanded in the following sense 

{w a i} : ={w a i> W a 2> (l)°3 } := { i (<)¥>,2> (<)*}| { 3 =0 ' for£=-l,0,l. 

(2.9) 

The co- vectors j ^ai }, dual to the vectors j ^a* }, are defined by the standard orthogonal 
relation 

< {e) a j , W 0i >= S{ , (2.10) 

where 5j is the Kronecker delta. The convected basis in the initial configuration is given 
by specifying t = in the spatial-centroidal-surface convected basis ^a* to obtain 

{{t)A L } := [ {i) Ai, {e) A 2 , } := { (QOi , w a 2 , W a 3}| t=Q , for£= -1,0,1. 

(2.11) 



12 

The basis j (t)A- }, dual to the initial-centroidal-surface convected basis { (e)Aj V, is de- 
fined similar to Eq.(2. 10). 

The membrane strain ^e, the transverse shear strain , and the bending strain 
measure (^p , which (as we will see later) are conjugate to the effective resultant membrane 
stress (t)^ 01 , the transverse shear ^q a , and the resultant couple (i)mP a , respectively, can 
be defined as follows 

:= (f)€o0 w a Q <g> {e) a , (2.12) 
(e)8 ■= (t)6 aii )a a , (2.13) 
(t)P (e)Pap (e)a a $ (<)o". (2.14) 

The components of the membrane strain (^e, the transverse shear ^)5 a , and the 
bending strain w p are given in the following relations, respectively, 

(3*0*1 = ^ ((0 a <*/3 - , (<)*a = {i)la - {tft% i (e)Pa0 = (<)Ka/J ~ , 

(2.15) 

where 

(<)«*o0 : = M^'to^i := (i)A a -( t) A , (2.16) 

are the components of the Riemannian metric tensors of layer (£), ^a a0 w a a ® ^a 
and (<)>l Q/ j (^)A Q ® {t)Ar, in the current configuration and in the initial configuration, 
respectively (Naghdi [1972], Marsden and Hughes [1983]). The shear strain measures, 
which are measures of how much the director {l) t and the director {e) t Q depart from the 
normal to the centroidal surfaces in the current configuration and in the initial configuration, 
respectively, are defined as 

{i)la = (t)a a ■ {e) t, ^7° = (/) A a • {e) t . (2.17) 

Finally, the current and the initial (nonsymmetric) director metric {e) K a(3 and {e) K° af3 for 
layer (£) are defined as 

(«)«<*/? := (*)a a • w t , {e) K° ap := (e) A a • {i) t 0>(3 . (2.18) 



13 



Using the above definitions, the stress power T c of geometrically-exact sandwich shells 
can now be written (Eq. (199) of Vu-Quoc et al. [1997]) as 



dA, (2.19) 



where (i)n al3 , (t)rh al3 and ^q 01 are the components of the weighted effective membrane 
force, the weighted resultant couple, and the weighted effective shear force, defined in 
(169), (174), (175) of Vu-Quoc et al. [1997] respectively. See the footnote at the beginning 
of this section for the meaning of the word "weighted." 
2.2.2.2. Power of assigned forces/couples 

Let n* denote the distributed assigned force on the centroidal surface of the refer- 
ence layer (0), and (^m* the distributed assigned moment on layer (£). On the boundary 
dA, we assume that the normal to the lateral surface of the shell domain in the material 
configuration is such that 

(f) I/ = (0)1/ = ( )^a E a . (2.20) 

The assigned force n* a and assigned couple ^)fn* a on the boundary dA are then defined 
such that 

n* = n* a {Q) u a , {e) fh* = {e) rh* a {Q) v a . (2.21) 

The power of the assigned forces and couples is written as follows (Vu-Quoc et al. [1997, 
(56)]) 

"Pa = f (n* -u + (t)m* ' (<)< ) dA 

+ I [n* a -u {Q) u a + £ (£) m* Q • {t) t {t) u a ) d(dA) . (2.22) 

d A V «— 1 / 

2.2.3. Constitutive Relations 

For layer {£), we employ the following constitutive relations between the above strain 
measures and the mentioned effective resultant forces/couples 

W n - / r2~W- W e 7<5> ( 2 - 23 ) 

1 -(it)V) 



14 



~0a {1)3 1 (?) E [t) Hi -Pa-yS . r9 9 ^ 

w to m = — f TzTW^ WPiS i 12.-24; 



12 



= (<)Jt {t)tl {t)A {1)0 



G inH mA al3 11)83 , (2.25) 



where 



{i)H := + w h~ , {e) A aP := {e) A a • (^A^ , (2.26) 

are the thickness of layer (£) of the sandwich shell, and the Riemannian metric tensor in 
the initial configuration for layer (£) of the sandwich shell. In Eqs.(2.23) to (2.25), m& is 

the Young's modulus, [i)G the shear modulus, the Poisson's ratio, and ^k s the shear 

5 

correction coefficient, for all layer (I). With mK s = -, Eq.(2.25) is the same as that given 

6 

in Naghdi [1972, p.587]. The elastic constant ^3 , with its component form given as 

W S*** = {e) u W A*> {£) A* + i (l - qu) qA* + {ei A(» {e) A^) , (2.27) 

is a fourth-order elasticity tensor. 

2.3. Weak Form and Linearization 

In this section, we construct the weak form of the equations of equilibrium obtained 
in Vu-Quoc et al. [1997], and linearize this nonlinear weak form, which plays an important 
role in finite element implementation. 

The shell layers are assumed to be inextensible in the thickness direction, and there 
is no drill degree of freedom (dof) for the directors (i.e., the directors are not rotating about 
themselves) considered. We have three translational dofs for displacement of the centroidal 
surface of the reference layer (0), and two rotational dofs for the directors of each layer. 
For a sandwich shell, the total number of dofs is nine (9). For the single-layer case, we 
refer the readers to Simo and Fox [1989] and Simo, Fox and Rifai [1990] for the details. 
2.3.1. Admissible Variations, Tangent Spaces 

The admissible variations to the deformation map <3> = ((o)V) (0)*, (1)*}. 

are denoted by 

5 $ := {5 {0) ip , 8 , 5 (0) t , 6 m t } . (2.28) 



15 



Let TS denote the tangent space formed by the Cartesian product of the tangent spaces 
T i S 2 (i.e., the tangent spaces to the sphere S 2 at w t £ S 2 , for £ = -1,0, 1). We write 



l 




(2.29) 



The space of admissible variations, denoted by B t (i.e., the tangent space to the current 
configuration B t ), at the current deformation <fr , is then defined as 

T$ B t := :^h->IR 3 x TS\6 {0) <p = 0ond<pA, 5 {i) t =0ond (<)t .4} , 

(2.30) 

where dip A and d ^ A represent the portions of the boundary dA where the essential 
boundary condition is imposed on <p and on (^)t , respectively. 
2.3.2. Weak Form of Equations of Equilibrium 

The weak form of the equations of equilibrium for sandwich shells is readily provided 
by the principle of virtual power, expressed by the following balance of power (Vu-Quoc 
et al. [1997]) 

y c = Pa , (2.31) 

where J* c and 7 a are the power of the contact forces/couples and the power of the as- 
signed forces/couples for the sandwich shell, respectively. 

It suffices to replace the time rates in the expressions for the powers y c and 7 a by 
the admissible variations S $ to obtain the weak form, which can now be written as 
Find * , such that 

G? c (*,<5*) = <?<,(**) , (2.32) 

for all admissible variations 6 <J> , where G c ( $ , S $ ) is the weak form of the contact 
forces/couples (or contact weak form for short), and G a (8 $ ) is the weak form of the 
assigned forces/couples (or assigned weak form for short). 



16 



2.3.3. Contact Weak Form 



The contact weak form of the contact power 7 C in component form is as follows: 

1 » M 



1 r n « • 1 

G c ( $ , 5 # ) = 51 / 5 <5 (f)«a/? + + 6 {1)1 a dA 



1 /■ ~ 1 

= H / [w na/3(5 W e a/? + {t)m al3 5 w p aP + (e^S^Sa^dA, (2.33) 

where (£)n Q/5 , (e)rn a0 and (^)<f* are the components of the weighted effective membrane 
force, the weighted resultant couple, and the weighted effective shear force, defined in Eqs. 
(169), (174), (175) of Vu-Quoc et al. [1997] respectively, whereas 5 ^)e a p = (1/2)5 {t)a a0 , 
5 (e)Pap — 5 (£)« Q /3 , and 5 ^S a = 5 (t)1a are respectively the variations of the strain mea- 
sures conjugate to the above weighted resultant tensors (Eq.(2.15)). The component form 
of the contact weak form is used in the computational formulation due to the constitu- 
tive laws Eqs. (2.23) to (2.25) that relate the weighted resultant tensors to their respective 
conjugate strains. 
2.3.4. Assigned Weak Form 

From the power 7 a of the assigned forces/couples in Eq.(2.22), we obtain at once 
the weak form of the assigned forces/couples 

G a {6& ) m J ^n* 'S (o)V> + W** ' 5 W* j dA 



f 1 

dA L «— 1 



d(dA) . (2.34) 



Remark 2. 1 . For the dimension of the assigned forces/couples n* and (i)fn* , we refer 
the readers to Vu-Quoc et al. [1997]. At the boundary dA of the sandwich shell, n* and 
(f)Tn* are decomposed as follows 



n* = n* a v a on d n A, 



w m* Q {e) u a on d m A , 



(2.35) 
(2.36) 



17 



where d n A and d (t)Tn A are the portions of the boundary where the assigned forces and 
the assigned couples are applied, respectively. I 

2.3.5. Linearization of Contact Weak Form 

To construct the linearization of the contact weak form Eq.(2.33) at a given configu- 
ration $ in the direction of an incremental tangent field: 

A $ := (A ( o)¥> , A ( _D« , A (0) t , A m t ,)er # 5, (2.37) 

we consider a one-parameter family of perturbed configurations 

£ h-» * e = ( (0 )^ e , (_l)* e > (0)*e . (1)** ) , (2.38) 

such that 

&e L = * < =A*. (2.39) 

The tangent contact weak form will be shown to be composed of a material tangent stiffness 
operator and a geometric tangent stiffness operator. The linearization of the contact weak 
form plays a central role in the computational procedure based on the Newton-Raphson 
method. 

2.3.5. 1 . Update of inextensible directors 

The following steps are used in the update of the inextensible layer directors. 

1. First, we must account for the assumption that the layer directors have no drilling 
dofs in their increments. The removal of the drilling dof is realized by nullifying the com- 
ponent A W T 3 of the material incremental directors A (/)T along the basis vector E 3 , 



that is 



A w r =A {e) T a E a , (2.40) 



where Greek indices take values in {1, 2}. Let the superscript fc on a tensor quantity 
denote the fcth iteration in the Newton-Raphson procedure. The spatial incremental director 
A ( t )t k for layer (£) is related to its material counter part A ^)T k through the orthogonal 



tensor w A fc as follows 



A(^«* = (() A*'A W Z* . (2.41) 



18 



If we do not make the distinction between tensors and their matrices of components, and 
the quantities in Eq.(2.41) in terms of matrices of component, then Eq.(2.41) can be written 

as 

A [e) t k = {e) A k A {e) T k , (2.42) 
where the matrix (£)A fc 6 R 3x2 is formed by the first two columns of the matrix (^A fc E 

ra3x3 



2. The spatial director is updated as follows 



{e) t k+1 :=exp . k 



sin 1 1 A^t k 



= cosH A w t* || w t k + " % " A {l) t k . (2.43) 

II A W* II 



3. The incremental director rotation matrix is obtained from the exponential map 

A w A fc :=exp 50(3) [ (t) 0] 

= cos|| {t) || 1 + sin|| {t) 9 || e + [l - cos (|| (t) ||)] e ® e , (2.44) 

where 1 is the identity tensor, [J and e are skew-symmetric tensors with and e 
as their associated axial vectors respectively, 

{t) := w t* xA w t* , e := F ^- J . (2.45) 

II II 

4. Update the rotation matrix of layer (£) 

w A fc+1 «A W A* W A* . (2.46) 

The above procedure is very important for the linearization of the weak form. 
2.3.5.2. Perturbed configuration 

Let A<I> := |A(o)V, A(_j)t, A( )t, A^tj G T<j> <S be the incremental field 
in the tangent space at the current configuration $ . The perturbed configuration along the 
increments in the tangent space is defined as follows 

*e = { (0)<P e . (-!)*« ' (0)*e , } , (2.47) 



19 



with 



(0)< 



{e) t £ := exp (<)t 



eA {l) t 



, for I = -1, 0, 1 , 



(2.48) 
(2.49) 



where 



r . I a n sm ll £ Aie)t || 

ex P w t [ eA M*j :=cos ll £A W* 11(0* + || £A{e)t || > 



(2.50) 



is the exponential map from T i S 2 to S 2 (see Simo and Fox [1989] for details). We now 



verify that Eqs.(2.48) and (2.49) satisfy (2.39). First, it is obvious from (2.48) and (2.49) 
that 



(0)V>« £=0 = (ojV 



e=0 



= {t) t for £=-1,0,1 



(2.51) 
(2.52) 



Second, by taking the directional derivative of (2.48) and (2.49), we obtain 
= A (0) v> , 



d 




de 


(o)¥> £ 


d 




de 





(2.53) 



£=0 



£ = 



-|| A {e) t ||sin(J| eA«)* ||j w t + || A (0 t ||cos [\\ eA (e) t 
= for/ = -1,0,1. 

Next, to linearize the strain measures, the following formulas are useful 



A w t 
A w t 



e=0 

(2.54) 



de 



(o)V> e 



£=0 



= A m t 



(2.55) 



e=0 



With the above results, we can now proceed to the linearization of the strain measures 
followed by the linearization of the contact weak form. 

Remark 2.2. The directional derivative of 8 ^t E along the direction of the increment 
A (^)i can be expressed in terms of the variation 8 y)t and the director as follows. It 



20 



is clearer if one thinks of the symbol 5 in the variation 5 ^t e as a derivative with respect to 
some variable, whereas the perturbation parameter e is a different variable. Since || \\ = 
1, it follows that mi • (e)t = 0, and we can define (qv := x such that 



i as w o> x w t , or 6 fat = <5 w x , 



(2.56) 



where ( f )U> and S ^0 are vectors that play the same role, one for the time derivative, while 
the other for the variation of (qt . For the perturbed director ^)t E , we have 



6 (i)t £ = 8(t)0 x ( f )t e , 



(2.57) 



with the same rate 8 ^0 as for 
Hence 

d 



de 



8 (t)t e 



5(i)0 x 



£ = 



de 



= <5 W x A m * 



£ = 



-(5 w t-A (£) t) ( t) t, far/ --1,0,1. (2.58) 



Since 



which is a result of (2.56) 2 and || w t ||= 1, and since • A w t =* 0. 



(2.59) 



2.3.5.3. Linearized strain measures 

Let ( (e)a E a i (<)7a/3 > ) t> e the strain measures corresponding to the perturbed 
configuration (2.47), as defined in (2.16)— (2.17), and let the incremental strain measures be 
defined as 

(A (fidof, , A W 7 a/J , A (QKan) := — ( ( t )a e a0 , ( £)7 Q /? . ) 



£ = 

for -1,0,1. 



(2.60) 



We obtain the following expressions 



A (e\CL 



(2.61) 



21 

A W 7a := (A (<) t • (e) <p <a + { t)t 'A {e) <p a ) , (2.62) 

&(QKa0 ■= •(/)*/» + (t)*P, a ' A w t l/? ) , for £=-1,0,1. (2.63) 

For £ = 1 and I — — 1 (i.e., the top layer and the bottom layer), we want to express the 
incremental strains A^a a 0, A^)j a , A^)K al3 in terms of the deformation map <fr := 
| ( o)¥?, (_i)t, (o)*, (i)*} and its increment A <E» :={A(o)¥?, A(_i)t, A (0) t, A (1) t}. 
This objective can be achieved by employing the constraints (2.4) in (2.63), and then we 
obtain 

A (1) a Q/3 = (A (0) v?, q ■ ( o)fP,0 + (0)¥>, a "&(o)<P,p) 

+ (0)h + (A (0 )VJ, a * ( O )t,0 + A( )<,/3 * ( )V,a + A(0)V,/3 ' (0)*a + (O)V 5 ,/? ^(o)i«) 
+ (A ( )V,a * + A {1 )tp « (0 )V?, a + A (0 )V> i/3 ' (l)t a + (0)^ * A (l)£,a ) 

+ (0)^ + (A(0)*,a • + (0)*,a *A(i)t^ + A(i)t >a • ( O )t,0 

+ (l)*a *A( )t,/?) + ((0)^ + ) (A(0)t, a • (O)t,0 + (0)*q "A( )t /?) 
+ ((!)/»-) (A 

A(l)7a = A (0 )¥>, Q • ( )t + ( )V,a •A(O)* + (0)/i + (A(0)t a * (1)* 

+ (o)*,a*A(i)*) + ( )/i _ (A ( i)t a »(i)t + ( i)t Q «A ( i)t) , (2.65) 

A(i)« a /J = A (0 )V,a • (1)*^ + (0)<P,a *A(l)*,/3 + (0)^ + (A (0)t a * (1)*,/? 

+ (o)*,a "A(i)*/j) + (A (1 )t a • (x)*^ + (i)i Q *A . (2.66) 
For layer (—1), we obtain 

A(_i)a Q/9 = A(o)V>, a ' (o)V,/3 + (0)¥>, a 'A(0)V^ 

- ( )/i~ (A (0 )¥> ia • (o)t,^ + A ( )i • ( )V ia + A ( )V i(3 • (0 )t Q + (o)¥>,/? * A (0) t „ ) 

- (A (0 )V3, q • (_!)* /J + A ( _ 1) t ) ^ • (0) <p Q + A (Q)^ • (_i)t a 



22 

+ (0)V,/3 •A(_l)t, a ) + (0)^" (0)* a ' (-1)* jU + (0)*,a "A (_!)<: (/ 3 

+ A ( )*,/3 * (-1)*,q + ( )t,/3 *A(_i)t iQ ) + ( ( o)/l _ ) (A ( )£, q • (0)*,/3 

+ (0)*,a •A(o)t^) + ( ( _i)/i + ) 2 (A ( _i)t a • { _i)t ija + (-!)*« • A (-i)*^) , (2.67) 
A (_i)7 Q = A ( o)V jQ • + ( )¥>, a * A ( _!)t - (0) /r (A (0) t, Q ■ (_i)t 
+ (0)*,a •A(-i) ,o • + (-yt^ »A(_i)t) , (2.68) 

A(_i)/C a /j == A (o)^ • (-l)*,/3 + (0)V, o ' A - (0)^~ (A (0)* a • {-1)*JS 

+ ( )*,a •A^t./s) - (A ( _i)t a • (_Dt l/3 + ( _i)* Q •A(_ 1) t i8 ) . (2.69) 

Remark 2.3. The above (2.63)-(2.68) are for the incremental strain measures for sandwich 
shell. To obtain the variation of these strain measures, we simply use the same relations 
with A replaced by 8. I 

2.3.5A Linearized contact weak form 

We now derive the linearization of the contact weak form, which requires the lin- 
earization of the resultant contact forces/couples. Substituting the one-parameter family 
of the perturbed configuration (2.47) into the static weak form (2.32), and then taking the 
directional derivative, we obtain 



DG C ($,5#)-A$ := 

ae 



(2.70) 

£=0 



The complete linearization of the contact weak form G c ( 3> , <$<!> ) can be divided into two 
parts, the material part and the geometric part. We will discuss these two parts in detail in 
this section. To make the derivation simpler, we express the contact weak form in matrix- 
operator format. 

2.3.6. Matrix-Operator Format of Contact Weak Form 

Let the material membrane force, shear force, and moment for layer (£) be defined as 
follows 



23 



(i)Jo v ' 



(2.71) 



M 



~11 — 22 — 12 

w m , (/) m , {e) m 



)'■ 



where is the Jacobian determinant in the material configuration evaluated at the cen- 
troidal surface of layer (£). 

We also define the director rotation matrix for layer (£) as follows 



A := 



and let 



A := , (£)*2 



(<)An (<)Ai2 (<)Ai3 

(£)A 2 l (£)A 2 2 (£)A 23 
(/)A3i (^)A 32 (i)A 33 



(£)An (<)Ai2 
(f)A21 (€)A 2 2 



(2.72) 



3x3 



(£)A3i (£)A 32 



(2.73) 



3x2 



which simply represents the first two columns in qqA . From here on, we will not main- 
tain a rigorous difference in notation between tensors and the matrices of their components. 
Thus, bold-face symbols are also used to designate the matrices of components of tensors 
with respect to the spatial basis { e x , e 2 , e 3 }. With this understanding in mind, the ma- 
trices of components of the deformation map of the reference layer (0) and of the director 
for layer {£) are written as follows 



(o)¥> := 



(o)¥> 
(o)<£ 2 



W 



t := 



(i)t 
(0 



L (0)P J 3x1 

while the variation and the increment of ( )V are written as 



t 2 
t 3 



(2.74) 



J 3x1 



^(o)V 1 
<5(o)</? 2 
<5(o)</> 3 



A( )¥> 



3x1 



A ( )V 2 
. A (o)¥> 3 



(2.75) 



3x1 



For the variation and the increment of the layer director {t)t , we need to account for the 
no-drilling dofs condition. The matrices of components of 5 and A are 



5 {e) t 2 
L 5 (i) t3 



A {e) t := 



3x1 



A ^t 1 
A {e) t 2 
A {e) t 3 



(2.76) 



3x1 



24 



The material counterparts of 8 ( ( )t and A y)t are respectively 8 W T and A W T , and are 
related to 8 and A by 

8 m t = {e) A 8 {e) T, A {( )t = w A A W T . (2.77) 

The no-drilling-dof condition imposed on 8 (<) T and A W T is written as follows 

8 m T -^3=0, and A (<) T • £ 3 - , (2.78) 

and thus if the matrices of components of 8 (£) T and of A W T (with respect to the material 
basis { Ej }) are defined as 



8 {e) T = 



5 ie) T* 



2x1 



A^T 1 

a w t 2 



2x1 



(2.79) 



then 



(2.80) 



Also 



8<f> := 



* (0)V 

*<-i)T 
S {0 )T 

L*d)T 



A$ := 



J 9x1 



A (0) r 

A(i)T 



9x1 



(2.81) 



We now will obtain the operator expression of the weak form for each of the three layers. 
Since layer (0) is the reference layer, to which the two outer layers are referred to, we begin 
with layer (0). From the membrane part of (2.33), the expression for 8 (o)Q a /3 similar to that 
of A (o) aa/3 in (2.63), and using the symmetry of the membrane forces (o)^ Q/3 > we obtain 



~a0 c 
o(0) n °(0) Oq/j 



-11 r i -22 r 

-(0)U (o)On + - (0)71 0(0)O 2 2 



+ 



\ ((0)" 12 5(0)Ol2 + (0)n 21 S (0)O21 ) 



d 



d 



= I ^T 5 (°)^ ] (o) n +1 (0)V>,2 ^77^(0) V J (0) 



n 



22 



+ ^fS^^WV + (0)v5^r5(0)V»l (o)" 12 . (2.82) 



25 



Introducing the following operator for the membrane action in layer (0) 

. d 



(o) B mm 



we can then rewrite (2.82) as 
1 



(o)V>,i 



t d 

(0)V,2 



t d , d 

(o)V,i + (o)V,2 



(2.83) 



3x3 



(0) 



n a0 8 m\a n a = 



(O)<ta0 = (0)3 o 



(0) 



B mm 8 ( )V 



(0) 



N 



(2.84) 



Similarly, we introduce the following operators related to the bending and shear actions in 
layer (0): 

d 



(0)Bbm 



(0)*1 



(0)Bbb 



t 9 



(2.85) 



3x3 



t 9 

(0)V,2 



d 



(2.86) 



d 



3x3 



*5 MM • 



(O)-Dsm 



(0)1 



^e 1 
^e 2 



(0)-Ds6 



(o)V»;i 



(2.87) 



2x3 



2x3 



Then following the similar procedure as described in (2.84), from (2.33), (2.63), (2.62), we 
obtain the operator format for the shear part and for the bending part of the weak form as 
follows 



- 1 



(o)<T<5(o)7a = (o)Jo (o)B am 5 (o)ip + (o)B sb 6( )t (o)Q , (2.88) 



(0) 



h a0 8 t 



(O)K a — (Q)j (o)-Bftm <5 (0)<fi + (0)B bb 8 ( )i 



(0) 



M 



(2.89) 



26 



The contact weak form (2.33) for layer (0) can now be written as 



(0) G C (*,<**) = 



/{ 

A 

+ 



t ~ 



(o) J5mm 8 {o)<P (o)N + (o)B sm 5 (o)<p + (o)B sb 5( )t (0 )Q 
(0)5^5(0)^ + ( )B bb 6 (0 )t] £ ( )M \ ( )j dA. (2.90) 



Remark 2.4. We refer to Simo and Fox [1989, eq. (6.25)], which is an expression similar 
to (2.90). I 

To obtain a simple representation for all three layers, we define the following gener- 
alized resultant force for layer (£) 



R := 



(e)Q 



M 



(2.91) 



8x1 



Recalling the relationship between 6 y)t and 8{f)T as given in (2.77), we combine the 
differential operators for membrane strain (2.83), for shear strain (2.86), and for curvatures 
(2.86), all for layer (0), into 



(o) 



B 



(0)°nm 3x3 03 X 3 3x3 
{0)B sm 2X 3 (0)B s f, 2X 3 
(0)Bbm 3x3 (o)Bbb 3X 3 



(2.92) 



8x12 



where A is the director rotation matrix for all layers defined as 

(-D A 



A := 



(i) J 



(2.93) 



12x9 



Then, the contact weak form (2.90) for layer (0) can be written concisely as 
( o)G c (*,<$#) = J {0) B6$* m R {0) ] dA. 



(2.94) 



For the membrane part of (2.33), the expression for S (_i)a Q/3 similar to that of A ( _i)a Q/? 
in (2.67), and using the symmetry of {-\)n al3 , we obtain 

~a8 c ^ —11 r 1 ~22 r 

-(_!)n d ( _i)a a/3 = - ( _i)n d ( -i)a n + - ( _ 1} n d ( _ 1) a 22 



27 



+ \ ((-i) n12 S (-i) a i2 + (_i)n 21 5 ( _i)a 2 i) 

<*(0)V,1 * (-1)^,1 - (-l)^ + <5 (-!)*,! * (-1)^,1 - ( )^~<5(o)*i * (-i)V 3 ,! 



+ [<* (0)^,2 * (-1)¥>,2 - (-1)^ + 5 (-1)*,2 * (-1)^,2 - (0)^ <5 (0)*,2 * (-1)V,2 

+ (_ 1} n 12 [<5 (0)^,1 * (-1)V,2 + 5 (0)V, 2 ' (-1)^,1 - * (-1)^,2 



(_l)/l + (J(_l)t2 * (-1)^,1 - (0)h 5 ( )*,i * (-1)V,2 ~ (0)/i <5(0)*,2 * (-1)V>,1 



(2.95) 



Upon introducing the following operator (-i)-B memb associated with the membrane action 
in layer (—1) 



(— l)-*-*memb • — 

. d 



(-*>¥> i 

(-^,2^1 



05x3 



(-1)^*1 



5 



. _ « d 
- {0 )h ( _ 1)¥>1 _ 







1x3 



(-1)^2^ 







1x3 







5x3 



(o)h~ (-1)11! lx3 

05x3 05 X 3 



(2.96) 



where the operator r-nlli is defined as 



A 9 



d 



(-1)11! := ( _ 1)V > 2 _ + i-d^ , 



(2.97) 



we can rewrite the membrane part for layer (-1) in (2.33) in a compact format as follows 

-(_i ) n Q/3 ^ ( _ 1) a a/3 = (_i)B memb <5$- . (2.98) 

From (2.68), we define the differential operator associated with the shear action in layer 
(—1) as follows 



(-i)B shear 



28 



3 x3 

. d 

dp 

03x3 



3 x3 



t ® 

(-D^,i - (-1)" (-1)* 



(o)h ( _i)t 



3x3 o 3x3 
d 







1x3 



(-1)^2 - (-1)* 



i 9 



dp 



(o)h (-i)t 



03x3 



0*1 
03x3 3x3 







1x3 



(2.99) 



Similarly, from (2.69), we define the differential operator associated with the bending ac- 
tion in layer (—1) as follows 

(-l)Bbend '■ = 



o 5x3 




05x3 




05x 3 






TT 9 


- (0)^ _ (-1)*, 1 


9^ J 






TT 9 


- (o)h~ (-i)t 2 


d ' 

dp 








- (0)/l~ (- 


1)114 







1x3 







1x3 







1x3 



A , 



(2.100) 



where the operators (_i)II/, for / = 2 , 3 , 4, are defined as 

(-1)112 := (-i)^i - (-i)^ + (-1)*, i i 

(-1)113 := (-i)¥>' 2 _ (-i)^ + (-1)*,2 i 

t d 



(2.101) 



(-i) n 4 : - (-1)^2 ^tt + (-i)*!i 



dp 



dp ' 



We can easily verify that the shear part and the bending part for layer (-1) in the weak 
form can be written in a compact format as follows 



(-1)9°" <*(-l)7a = (-l)B s hear<5* * (-l)-R, 



(2.102) 



29 



(_l ) m Q/3 ^(_i ) /C Q/3 = { -i)B hend 8$ ' (-l)-R 



Now let the combined differential operator for layer (—1) be 



(-l)-B : — (-l)-Omemb T (-1)-° shear T (-l)-Dbend 



-Bmemb + f-ll-Bshear + t-l)By 



From (2.96), (2.99), and (2.100), we obtain 



B = 



j 9 



(-DV,i 
t d 

(-i)II, 
t 5 

(-«* ae" 



(-1)^,2 



(-»« ^2~ 



- (-i)* 

(-1)^2 - (-1)* 



(-D /i+ (-l) n i 
5 



(-l) n 2 



(-l) n 3 



g 



(-1)114 



TT 5 TT 5 

(-1)112^- +(-!) n 3^T 



We thus obtain the following compact expression 



1 



( o)/r (-i)v; 2 ^i- u ix3 

- (Q)h~ (-i)Hi lx3 
5 



- (o)h (-i)t l 



(o)h { -i)t 



(0)h (-1)*,2^72 Ulx3 



d 



(2.103) 



(2.104) 



0, 







1x3 







1x3 



t d 

(0 )ft (-i)t,i ^-f lx3 



0, 



(o)^ (-i)n 4 o lx3 



(2.105) 



- ( _ 1 )n Q/3 5 ( _ 1) a Q/3 + ( _ 1) 9 Q 5 ( _ 1) 7 Q + ( _ 1) m^5 ( _ 1) « Q/ 3 = ( _ 1} £ 5 $ • { _ 1} R , (2.106) 



and the contact weak form of layer (—1) as 



(2.107) 



For the top layer (1), similar to the definition of (-i)J3 for layer (—1), we define 



(i) 



B := 



d 



d 



(DVfi °ix3 (o) h+ tiMi 7*71 



d 



d 

(i)^2^2 01x3 m h+ 

(0)h + (1)11! 



(1)11! 0i x3 

. d 



(i) 1 



d£ 2 



0lx3 (0) h + (1) t* 



a 







1x3 (0) 



0^ 



0^ 

J? 

d 



T^TT ° lx 3 (o)^ + (l)*4 



(1)*,2 







df 2 

(i)Il 6 lx3 



1x3 (0) h+ {1) t- 2 



t d 



(o)h + (i)II 6 



w 71 (i)^Ii ^71 



(i)*"(i) n i 

(l) n 2 
(l) n 3 

n 9 



d 



(1)114 



d 







(i) n 5^FT +(i) n 4 



d 



30 



(2.108) 



where the operator (1)11/, for 1=1, 6, are defined as follows 

tt t & t ® 

m lli := ( i)¥> )2 ^j- + (i)V>,i , 



(l) n 2 



(l) n 3 



M<P,i + (i)h (i)t — , 

(1)^2 + d) <£ ^" . 

(1)114 := (1)^1 + (i)*,! , 

(XjIIg := (i)V>f 2 + (1)*,2 , 

> t 3 . d 



n fi := 



(1)116 



ae 2 



(2.109) 



31 



The contact weak form of layer (1) is then 

(1) G C (*,<$*) = J (1) B6*- (1) R {1) j dA. (2.110) 

A 

2.3.6.1. Material tangent operator 

The material part of the tangent stiffness operator, denoted by DmG*A$ , arises as 
a result of linearizing the resultant forces/couples at a fixed configuration. We now treat 
each layer separately, as we did for sandwich beams in Vu-Quoc and Deng [1995]. Here, 
we only consider hyperelastic materials. Let tp be the energy function of the shell. We have 
the following constitutive relation (Simo and Fox [1989]) 

dip _ dip _ dip 

(e)n a0 = (i)p , {l) q a = {t)P « , , {i)m a0 = {t) p- — — . (2.111) 

O(e)£a0 O {l)Oa O (t) Pa0 

For each layer, we have 

2> W R«A$ = (i)C( f )BA$ , (2.112) 
where the tensor (^C of elastic moduli is given below 




V 
CN 



<0 



% 2 



(N 

CO 



n T co 



•^crT 

cn 

CO H 

Qi 

co" 



-9co" 

%> 
els' 



•&CO 
CO - 



CO 

« CN 
^ -I 

to 

co" 



CO 



Q. 

CO - 



CN 

CO 



CO 



CO 



CN 

«0 



c6" 



co" 
cn 

CN 
54. 

els' 



CO' 















•3 


CO" 








CN 






















CO 





-9- — 

cn CO 



CO - 
CO" 



cn 
CO 







iH 


CO" 




V 

ft 


tN 
CN 


CN 

CO 


CN 


Ci. 




a 


co" 







-^.CO 
CN H 



CN M 
"2 



-3.CO 

CN CN 



CN 

CO 



CN 

CO 



CO 

co" 



•3- co 

CM 
* 



co" 



-5- CO 

CN 

CO 



-=>.co 

CN " 

05 £ 



3 

CN 

CO 



* CO - 
CO c-o 



CN 

CO 



CO 



CO 

CN 
CN 

^> 

co" 



■3- co 

CN 

co 2 



-9- 

CN 

CO 



CO 



CO 



CO 



CO 



-5H - 
CO 



CO 



cn 



«0 
c$* 



9 

CN 

CO 



CO 

tN 
tN 

c6" 



CN 
CO 



CO 

CN 

co" 



CN 

CO 



-3co 

tN 

CO M 
to 

co" 



•3 - 

IN co 



CN 

cb 



-3). 

CN 

CO 



CO 
co" 



CO 

CN 
C4 

^* 

co" 



%3- 

CN 

CO 



CO 

CN 

s 

CO" 



"9- co 

CN ^ 

CO 2 
cu 



■3 =• 

CN CO 



-9- ~ 

CN CO 



CO 



CN 
CO 



CN 



CO 



tN 

CO 



o 
l<5- 



33 



For hyperelastic materials, the elastic moduli for the membrane, shear, and bending 
actions for layer (£) are given as in (2.23)-(2.25). The matrix of the (tangent) elastic moduli 
in this case takes the following form 



(*)C := 



sym. 



with 



(£)C m := 



{e) c% 



(t)C s := 



sym. 



/->33 



03x2 


03x3 


(e)C s 


02x3 












(e)C b := 


3x3 





(2.114) 



8x8 



■>11 



sym. 







, x/ n '22 









3x3 



(2.115) 



2x2 



The coefficients of the above matrices are given below, for the membrane action: 

(e)E {e) H 



^13 



/o22 



r<23 



/o33 



1 - («)f) 
W E W H 



2 W^ 11 M^ 11 . 



»'»i4 11 W A«+(l-(o«') W ^ 1, W ^ U ]. 



1 - (w^) 

1- ((/)»/) 



2 W^ 1 (*) 



/l22 4 22 
2 W A (i) A 



(2.116) 



22 /i 12 



2 tt)A tf)*- , 



1 



- w 



W A + — - — {e) A (e) 



12 „ A n 



34 



For the bending action: 

d)E {e 



WW 



C 13 



12 (l - 

{i)E {l 



12 (l - 



CP 



12 (l- 



12(1 - 

(i)E {l 



CP 



W^b 



12(1 - 



12(1 - 

For the shearing action: 



# 3 



A n A 11 



0" 

tf 3 



# 3 



H 3 



H 3 



H 3 



(0 



v {e) A n {e) A 22 + (l - m v) {l) A 12 {e) A 12 } , 



All Al2 

{I) A ( t )A , 



(t)A 22 {e) A 22 , (2.117) 



A 22 4 12 



l ~ M V 4" ./I22 , 1 + (t) V 4 12 Al3 

{t)A (e)A H f^/l f^A 



W^s 11 = W«s (<)A U , 

(()Cl 2 — [i) K » {t)G (i)H ( ( )A 12 , 



C 22 = (t)K s (£)G (t)H (t)A 22 



The tangent material stiffness operator for layer (£) is thus 
V M[i) G c (#,J#)-A# = I [ W B*$« (<) C W BA$ 
2.3.6.2. Geometric tangent operator 



(2.118) 



dA. 



(2.119) 



The geometric part of the tangent stiffness operator, denoted by D G G*A&, arises 
from the linearization of the geometric part of the contact weak form, while keeping the 
material resultant forces/couples constant. We now treat each of the three layers separately, 
as we did in dealing with the material stiffness operator. 

Remark 2.5. It is noted that while the principal kinematic unknowns are mtp and 
[fit, for I = -1,0, 1, the computational kinematic unknowns are $)(p, , for I = 



35 



-1,0, 1, where mO represent the rotation vectors that rotates E$ to {e) t at the current 
state at time t. It is important to note that {i) 9 does not represent the time history of the 
motion of the director {l) t , but only relates the directors between the material configuration 
and the current configuration. 

In the linearization procedures, the primary variables (i.e., the variations to be held 
constant in the linearization process) are S {t) instead of 5 {e) t , which we will explain in 
Remark 2.6. We recall that 

|| {g) t ||= 1 =>6 {t) := [^t x 5 {e) t, or 6 {i) t = 5 {t) x (t) t . (2.120) 

we thus obtain the increment of the 5 from (2.58) as 

A(<S ( /)i) =5 {e) 0xA {e) t = ( w l x5 w t)xA {e) t = - (A (<) t -5 w t) (e) t , (2.121) 



I 

Remark 2.6. From (4.19) and (4.12b) of Simo and Fox [1989], (84) 3 and (156) of 
Vu-Quoc et al. [1997], the equation of balance of angular momentum for a single-layer 



and also 





(A {e) t'S {e) t) {t) t a . (2.122) 



shell is 




Jt 



(2.123) 



Alternatively, it also can be written in the following form 



1 




(2.124) 



Jt 



Since II t 11= 1, we differentiate it twice to obtain 



t ' t 



l=>ft=0=$>t't + t't = 0. 



(2.125) 



36 



Thus, t _L t , let « be define as 



w := t x t . (2.126) 



We obtain 



u = txt + txt=txt. (2.127) 



Since 



(a? x t)^t*[w,t,Jt]*w»(t X<5*) « ut'69 , (2.128) 

w x t = x V J X t (*•*) t - (* •*) * = *+ II * f * . (2.129) 
and also since t *5t = 0, we obtain 

(w x t) -6t = , (2.130) 

so that 

Tp'i'St = 1 p u '89 . (2.131) 

In the above equation, I p u> corresponds to the inertia term on the right hand side of 
(2.123) (i.e., the balance of angular momentum in terms of the weighted resultant moment 
fh a ), whereas I p u corresponds to the inertia term of (2. 1 24) (as a result of (2. 1 27)) (i.e., 
the balance of angular momentum in terms of the physical resultant moment m a ). We 
now point out that the reason to use 8 , instead of 5 1 , as primary variable: The use 5 9 
is more convenient. Note that 

i ^ w x t . (2.132) 
Since t • t = and (2.126), we obtain 

t-wxi, (2.133) 
u> x t = ijj x (uj x t] = (wt\ w - (wu>) t = - || uj || 2 l»t , (2.134) 



37 



Making the derivative of t = u> x t , we obtain 



** • * • n2 

t — IjJ X £ -f U> X t = u x t — II UJ 11 1 • I 



(2.135) 
I 



For the reference layer (0), in (2.94), we hold the resultant forces/couples ( )-R fixed, 
and linearize the geometric part, that is, we are finding the expression 



[P( (0 )B5*) *A$] '(o)R , 



(2.136) 



in operator form, where the operator ( 0) B was given in (2.92). From (2.92) and (2.81), 
(2.136) can be expressed as 

X>( (0 )B(J*) • A* •«,)•** = {v[ {0) B mm 6 {0) (p] *A (0 )#}'(o)iV 



[i] 



+ 



[V [(o)B m S (0 )V + (o)£ s6 <5 (0 )T] •A(o)*} • (0 )Q 



[2] 



+ {D [wBbmSwfp + i0) B bb S [0) T] .A (0) *}-(o)M, (2.137) 



[3] 

where the differential operator (o)B m , (o)B sm , $)B*ni (o)B sb ,and ( )B bb were given 
in (2.83), (2.86), and (2.87), respectively. We now proceed to give a detailed expression for 
part [1] in (2.137). From (2.83), we obtain 



[1] = 



A(o)V'i (o)V,2 + A(o)^'i ^(o)V,i . 



(0)TI 

—22 

(0)? ? 

(o)» 12 



[A ( o)^! •5ffl)(fi 1 (o)n 11 + A (0 )Vj *<5(0)V,2 (o)« : 
+ { A (o) ( P\ '^(0)^,2 + A (o)^i (o)V,i ) (o)^ 12 



xx22 



(2.138) 



38 



We define the geometric differential operator for layer (0) as 

d 



(0) 



T := 







03x3 3x3 



o 



03x3 3x3 
03x3 3x3 1 3 

03x3 3x3 1 3 



3X 3 3x3 



Is 



3 ^3x3 



d 



d 







3x3 



o 3x3 



A 



12x9 



(2.139) 



15x12 



and the tangent geometric moduli for the membrane action in layer (£), for £ = -1, 0, 1, as 

(2.140) 



{e) n lL 1 3 {e) n LZ 1 3 3x9 
W" 12 Is (e)n 22 1 3 3x9 

0g x3 9x3 9x9 



It is easy to verify that 



[1} = {v[ {0) B m 5 {0) <p} •A (0) #} « (0) JV = (0) Ti$' (0) JfJJ (0) TA$ . (2.141) 



Now, for Part [2] in (2.137), from (2.86), we obtain 

[2] = \v [(o)B sm 6 {0 )<P + (o)B sb 5 {0 )t »A(o)$ } • (0 )Q 

= [A ( oy**-<J ( o>V>,i (o)? 1 + A (0 )i ( *6 i0) <p t2 (0) 9 2 +A (0) ¥>; 1 -5 {0) t ^q 1 

~ (<Q)V>fi (o)*) (^(ojt »A(o)t) (o)g 1 + A (o)^ «<5( )t (o)<7 2 

- ((0)^2 (o)*) (<5(o)* *A (0) t) ( o)g 2 ] • (2.142) 

Remark 2.7. The matrix (0) T in (2. 1 39) has five rows of submatrices and four columns 
of submatrices. The four columns of submatrices correspond to {0) <p, { _ 1} t , {0 )t , mt ac- 
cording to the ordering in 5 $ (See (2.81)), and 5 {e) t = {e) A 6 {l) T (See (2.80)). I 



39 



Let the tangent geometric moduli for the shearing action in layer (£), for I = — 1, 0, 1, be 
defined as follows 



Tfl2 



3 x3 


3x3 


I3 





3 x3 


3 x3 


(£)Q J-3 







WQ 2 Is 


- (09° (<)7a I3 





06x3 


06x3 


06x3 






(2.143) 



It can be verified that (2.142) can be written as 

[2] = {p[(o)B sm £ (0 )¥> + (o)B sb 5 {Q) t] 'A(o)*} • (0 )Q 



(o)T^' ( o)4 2 (o)TfA$ 



(2.144) 



For part [3] in (2.137), from (2.87), we obtain 
[3] = {V [ 

(0)-B6m^(0)V + (0)-B 66 (5(o)tj 'A( )f }'(0)M 

= [( A (o)*?i ^(o)V,i +A( )V»; 1 *^(o)*i + (0)¥>a •A^<o)*,i) (o)^ U 

+ (A ( )t* 2 '(5(0)^,2 + A (0)^2* 5 (0)*, 2 + (0)^2 * A5 (0)* 2 ) (0)^ 
+ ( A (0)*fl (0) V,2 + A (0)^2 (0)V,1 + A (0)^i *6 (0 )t,2 

+A (0 )vJ 2 ^(o)*,i + (0)V»*i •A<5( )t,2 + (o)^2 *M m t,i) (o)m 12 } ■ (2.145) 
Let the tangent geometric moduli for the bending action in layer (I), for £ — —1, 0, 1, be 



defined as follows 



3 x3 


3x3 


3x3 


(e)rn 11 1 3 


w m 12 1 3 


o 3x3 


o 3x3 


o 3x3 


w m 12 1 3 


(e) m 22 1 3 


o 3x3 


o 3x3 




- (e)m la (/)7a 13 


- w m 2a W 7 Q 1 3 


(t)m n 1 3 


{i)rn n 1 3 


- (e)rn la {( )la h 


3x3 


o 3x3 


{e) m 21 1 3 


(t) m 22 1 3 


- (i)rn 2a W 7 a 1 3 


o 3x3 


o 3x3 













(2.146) 



Using Remark (2.5), we can verify that 

[3] = |X> [^BbmS ( 0) (f + 



(0)^66^(0)* 



'A ( o)^} 



(0) 



M 



(2.147) 



40 



Let the geometric stiffness moduli for each of the three layers be defined as 



r ~ii 

(0 n 


J-3 


=12 i 
(0 n 13 




. /rl 1 


(0 m X 3 


(0 m L 9 


{Qtl 


1 3 


(£)n 22 1 3 




—2 1 


w m 12 1 3 


(e) m 22 1 3 




la 


(0<7 x 3 






- w m lQ (£)% 1 3 


- (0^" 2Q (<)7a 13 


(i)rn n 


u 


w m 12 1 3 


™ ic* 
- (0 m 


(07a 1 3 


03x3 


03x3 


— 21 

. (t)m 


h 


{e) m 22 1 3 


- (f)m 


(07a 13 


03x3 


3 x3 



(2.148) 



where 



{e) C := - (07a + (e)m a/3 «)««„) . (2.149) 

Adding up (2.141), (2.144), and (2.147), we obtain the following expression for (2.137) 



{0) R = (0 )T<5*»JC£ ( o)TA$ . 



(2.150) 



[d( (0 )B(5$) 'A$ 
The tangent geometric stiffness operator for layer (0) can thus be written as 

Do<oA (♦.,**) -A» = | [(fl)T^- ( o)^(o)TA$] ( o)7o^- (2-151) 

A 

For the bottom layer (—1), in (2.107), we now hold the resultant forces/couples 
(-i)-R fixed, and linearize the geometric part, that is, we find the expression for 



»A#] •(-i)Jl 



(2.152) 



41 



in operator form, where the operator (-i)-B was given in (2.105). We define the two 
differential operators (-i)Ti , and (-i)T 2 for the bottom layer (—1) as follows 



(-i)Ti := 



V 1 


-(-1) 








d 






- (o)h~ 


i 3 9 


3 x3 




-is 




3 x3 


03x3 








03x3 


03x3 




1 a 




03x3 




" o 3x3 


13 


3 x3 


3 x3 " 




3x3 


l3 ae 


3 x3 


3 x3 




3 x3 


3 ae 2 


03x3 


3x3 


T 2 := 


03x3 


03x3 


13 


3X 3 




3 x3 


3 x3 




03x3 




03x3 


03x3 


13 o e 


03x3 



3 x3 

3 x3 
03x3 
3 x3 



12x9 j 



15x12 



(2.153) 



12x9 



(2.154) 



18x12 



In addition to the tangent geometric moduli (-\)K l G that corresponds to the bottom layer 
(-1) as an independent single-layer shell, we have also the tangent geometric moduli 
(-i)-Kg tnat comes from the coupling between the bottom layer (-1) and the reference 
layer (0). The tangent geometric moduli {-i)K% for layer (-1) can be written as follows 



42 

(-l)-K'l 13 {-\)K 2 lz (-1)^3 13 

{-1)^2 I3 O3X3 03x3 0g x g 

(_l)i^3 1 3 3X 3 3X 3 



(-1)^4 1 3 


(-l)-^5 13 


(-i)K e 1 3 


(-1)^5 I3 


03x3 


3 x3 


(-i)K & 1 3 


3 x3 


3 x3 



(2.155) 

The parameters in the above moduli matrix ^i)Kq are 

: = {-i)h + ((-D^ 11 (-1)^1 + (-i)^ 12 (-1M2) (-i)*.i 

+ (-i)h + ((-i)n 22 (-i)vfa + (-i)" 12 (-i)Va) (-i)*,2 

+ ((-i)m 11 (_i)t, 1 • 1 + (-i)m 22 2 * (-1)^,2) 

+2(_!)/i + (-i)m 12 • (-i)*,2 1 

(_D^ 2 := ((-i)« n (-1)^1 + (-ip 12 (~1)<P%) (-i)t + (-i)^ + , 

(-l)#3 — ((-i)^ 22 (-1)^,2 + (-D« 12 (-1)^1 ) (-1)* + (-D^ + (-i)Q 2 . 

(-i)K 4 := (0) /r ((-i)« n (-i)Vj + (-i)" 12 (-1)^2) (o)*,i 

+ (0)h" ((-i)« 22 (-i)<p% + (-i)^ 12 (-l)V'l) (0)*,2 

+ (0)/l~ ((-l)? 1 (0)*, 1 + (-l)Q 2 (0)*,2) ' (-1)* 

+ ( )/i~ ((-i)fn 11 • (o)t,i + (-i)"^ 22 (-i)*2 ' (0)^,2) 

+ (o)^~ (-i)™ 12 ((-i)*,2 * (0)*,i + (-l)*l * (0)^,2) , 
{-i) K s := (o)h~ ((-i)n 11 (_!)V»fi + (-i)™ 12 (-1)^2) (0)* 

+ (0)^" (-i)^ 1 (0)* * (-1)* 



43 



+ ( )/T ((-i)rn 11 (_i)tj • (0 )t + (-i)m 12 ( _i)t 2 * (o)*J . 
( _i)K 6 := { )h~ ((-i)n 22 (_i)^2 + (-i)" 12 (-D^i ) (o)* 
+ (o)h~ (-i)Q^ (-1)* ' (o)t 

+ (o)h~ ((-i)rn 22 (-i)*,2 * (0)* + (-i)"^ 12 (-i)*.i * (o)*) • 
The tangent geometric stiffness operator for layer (-1) can thus be written as 

V G{ -i)G c (#,*$)«A# = J [ (il )T 1 5¥«(_i ) ICj; (-dTjA^ 

.4 

+ { _ 1) T 2 5$- ( _ 1)J ft: 2 7 ( _dT 2 a*] (-dJo^- 

For the top layer (1), in (2. 1 10), we now try to obtain the expression 



(2.156) 



(2.157) 



(2.158) 



[P( (1) B<5*) -A* 

in operator form, where the operator (i)B was given in (2.108). Similar to (2.157), we 
obtain the following tangent geometric stiffness operator for layer (1) 

V G[l) G c (■#,*•) -A$ = 

/ 



■ x {T l 6*' {x) K l G ( i)TiAl + (1) T 2 <5*' ( i)^G (i)T 2 A*| (1) j dA, 



(2.159) 



where 



(i) T i 



d 

d 



o 



3x3 (0)^ + J-3 



d 







3x3 (0)^ + J-3 



ae 1 



(i)^ 13 



d 



(i)h 1 3 







9x9 



d 

i 3 

d 



A 12x9 . (2.160) 



15x12 



44 



(1)T 2 := 



03x3 03x3 







3x3 



03x3 3x3 1 3 







3x3 



d 
d 

03x3 3x3 ^3^2" ^ 3x3 



03x3 3x3 

03x3 3x3 3x3 1 3 



03x3 I3 

d 



0,x3 



3x3 u 3x3 3X 3 I3 



d 



12x9 



18x12 













1 3 


(1)^2 I3 


(1)^3 h 




1 3 


3 x3 


3X 3 


(i)K 3 


1 3 


3 x3 


03x3 







9x9 







9x9 





1 3 


(1)^5 I3 


h 


(i)K s 


1 3 


3 x3 


3 x3 


(i)K e 


I3 


3 x3 


3 x3 



(2.161) 



18x18 



(2.162) 



The parameters in the matrix ix)K 2 c are defined as follows 



~(o)h + ((i)n 11 (i)(p\ + (1) n 12 (i)^) (0) t, 

~(0)h + ((i)n 22 (1) v>f 2 + (ijn 12 (1)^1) (o)*2 

~(o)h + ((i)^ 1 ( o)i, 1 + (0 )i 2 ) * (i)* 

~(0)h + ((ijm 11 {!>*,! • ( o)t 1 + ( i)m 22 (1) i 2 ■ (0 )t 2 ) 



45 



-(o)h + (i)rh 12 ((i)*,2 * (o)*,i + (i)*i * (o)*,2 ) ) 
(i)K 2 := ~(o)h + ((i)n n ( i)^j + ( i)n 12 (i)<^ 2 ) (0)* - (o)h + (i)? 1 (i)t • {0) t 
~(0)h + ((i)m n ( i)t,i • (0 )t + (i)m 12 (1) t 2 ' (o)*) ■ 

(1)^3 := ~(o)h + ((l)" 22 (i)Vf 2 + 0)™ 12 (i)^fl) (0)* ~ (o)h + (i)9 2 (l)* * (o)* 
- {Q )h + ((i)m 22 ( i)t 2 * (o)* + (i)^ 12 • (0 )t) , 

(i)^4 := (d)" 11 m<P*i + (i)" 12 (1)^2) - W h ~ ((D^ 22 (D^2 

+ (i)« 12 (i)<P*i) (i)*2 - (1)^" ((l)™" (i)*i •«*,! 
+ (i)"^ 22 (i)*2 ■ (i)*,2 - 2 (1) rn 12 1 • (iy*,a) , 

(1) ^ 5 := ~m h ~ ((I)"" WVfi + (!)" 12 (1)^2) (i)* ~ W h ~ (i)^ ' 

m Kt ■= -(o) h ~ ((i)^ 22 (1)^2 + (i)^ 12 (i)Vfi ) (i)* - (i)^" (1)9* • (2.163) 

Remark 2.8. Even though (\)K l G has the same form as (o)K G (i.e., the operator 
of a stand-alone layer (1)), the operator (1) Yj has the coupling terms in the submatrices 
(1,3), (1,4), (2,3), (2,4). In of (2.160), there are five rows of submatrices and four 

columns of submatrices. The four columns correspond to ( )V?, (-1)*, (o)*» (l)*- The 
coupling terms in (1 ) Y x , when multiplied with A ( )t and A { i)t in ( A A <*> ), only affect 
the submatrices (1,1), (1,2), (2,1), (2,2) related to the membrane forces {e) h n , w n 12 , 
(t)n 21 , (e)n 22 . The reason is the offset of layer (1) with respect to reference surface, which 
is the centroidal surface of layer (0). 

Another way to understand (i)Yi (or the meaning of the coupling terms in (1) Yj , 
i.e., the difference between (0 ) Y and (1) Yi ) is to think of layer (1) as a stand-alone layer, 
initially at the same location as that of layer (0). Then (i)Ti would be similar to ( ) Y 
(note the difference between the column corresponding to in (i)Yi and the column 
corresponding to ( )t in (0) Y ). Now shift layer (1) to the top of layer (0); then the mem- 
brane forces in layer (1) must generate some additional moments. The coupling terms in 



46 



M T, Play the role of lever arms. On the other hand, in (1) K 2 G , we have all the coupling 

I 

terms, but not in (i)"T 2 . 

2.4. Numerical Examples for Statics of Sandwich Shells 
The finite element formulation for the statics of geometrically-exact sandwich shells 
presented in the previous sections has been implemented in the Finite Element Analysis 
Program (FEAP), developed by R.L. Taylor (Zienkiewicz and Taylor [1989]), and run on a 
DEC ALPHA with the DEC UNIX V3.2D-1 operating system. Linear finite-element basis 
functions are used in the examples in this section. To avoid shear locking, selective reduced 
integration is used to evaluate the shear part of the tangent material stiffness matrix {l) K M 
and the tangent geometric stiffness matrix m K and also the shear part of the residual 
force matrix, whereas the bending part and the membrane part of the tangent stiffness 
matrix and of the tangent residual force matrix are evaluated using full integration. 

To identify the correctness of the present theory and the related coding, we tested 
several examples of sandwich plate with different aspect ratio. The aspect ratio is defined 
as A := i . W ■ T , where (L, W, T) designate the length, width, and thickness of the 
sandwich plate, respectively. For the bending stiffness and membrane stiffness, we use 
the full 2 x 2 integration points, whereas for the shear stiffness, we use the reduced 1 x 1 
integration point. 

Firstly we tested the bending of a sandwich plate with three identical layers employ- 
ing 90 four-node quadrilateral elements. The plate is clamped at the edge = 0, and is 
free at all the other edges (See Figure 2.2). The vertical displacements and the tip rotations 
are compared with the theoretical results and with the results obtained from the single- 
layer theory. The results, with a maximum error less than 0.4%, also show the ability of 
our sandwich shell elements to model the anticlastic curvature. 

We also consider the Cook problem with only the core layer using 100 sandwich shell 
elements. The results agree well with those of single-layer shells (Rifai [1993, p. 173]). 

To demonstrate the ability of the formulation to capture large rotations and displace- 



47 



e 3 = E 




*12 









h < 




/i i 



12 

^ (0)^ 

)<-l) 



m 



.12 



Figure 2.2. Sandwich shell with three identical layers 
ments, we tested the torsion of a cantilever plate with three layers using 20 sandwich shell 
elements (Figure 2.3). To justify the computed results, we use the theoretical rotation of 
the torsion of an elastic bar given by 6 = TL/GJ k as a basis for comparison, 3 where L 
is the length of the plate, G is the shear stiffness and J k is the polar moment of inertia of 
the plate cross section about the centroidal axis along the length of the plate. Since the 
direction of the concentrated forces remains fixed along the £ 3 axis, the resulting couple T 
generated by these forces decreases with the twisting of the bar (See Figure 2.4), as the dis- 
tance d between these forces becomes smaller. Thus to make the comparison meaningful, 
we use the final value of the couple at the last time step as the torque T used to compute 
the theoretical rotation 9. The difference between the theoretical results and our FE results 
on the twisted angle is 9%. 




Figure 2.3. Torsion of a cantilever plate. 



3 Roark and Young [1975, p.290] presented the formulation to calculate the angle of twist for beam with 
solid rectangular sectio°n, which gives very close results when compared to the above formulation « 0.2%). 



48 



£ 3 




d 



\ 



F 



e 



0.2 



Figure 2.4. Force couple, at the tip of the plate, generated by a couple of concentrated 
forces. 

2.4. 1 . Roll-down Maneuver of a Sandwich Plate 

We now consider the roll-down maneuver of sandwich plates. First, we tested the 
sandwich shell having only the core layer using 10 elements. Comparing to the theoretical 
deformed shape (i.e., a full cylinder), the relative error in the tip displacements is 0.4% 
in the f 1 direction and 0.005% in the £ 3 direction. The displacements obtained with the 
sandwich shell code are exactly the same as those with the single-layer shell code. We also 
tested the roll-down of a sandwich plate with only one outer layer. We still obtained good 
results even with the slower rate of convergence. 

Next, we consider the sandwich plates with three identical layers. The material prop- 
erties and geometric properties are chosen as follows: 



= 1.2 x 10 7 , w i/=0.0, w « s = 0.75, w h = 0.033333, for £=-1,0,1. 



where , , ^k s are the Young's modulus, the Poisson ratio, and the shear correction 
coefficient of layer (£), respectively. The geometrical dimensions of the plate are length 
L = 10, width W = 0.1. 

At first, we use 10 uniformly distributed sandwich shell elements to discretize the 



(2.164) 



49 



sandwich plate. The computed tip displacements at the end of the last loading step are re- 
ported in Table 2. 1 . The computed displacement u 1 differs from the exact solution by 7.5%. 
In the first loading step, convergence is achieved after 10 iterations; in the last loading step, 
convergence is achieved after 1 1 iterations. Then we use 20 uniformly distributed sandwich 
shell elements to discretize the sandwich plate. The computed tip displacements at the free 
edge are reported in Table 2.1, where it can be seen that the displacement u 1 is closer to the 
exact solution of (-10) when the plate becomes a full cylinder. The relative error in u 1 is 
now 1%. Convergence is obtained after 9 iterations in both the first loading step and in the 
last loading step. Finally, the computed displacements at the free edge using 40 uniformly 
distributed elements shown in Table 2.1 are clearly closer to the exact solution, in which 
the value of u 1 should be (—10), and the value of u 3 should be zero. The relative error in 
the displacement u 1 is now 0.53%. Convergence is obtained after 10 iterations in the first 
loading step, and 11 iterations in the last loading step. 

Table 2.1. Roll-down of a sandwich plate with identical layers: Displacements of a corner 
of the free edge. 



Disp. 


10 elements 


20 elements 


40 elements 


v> 


-9.25472 


-9.81873 


-9.94730 


u 2 


-3.08237 x 10" 11 


-2.15727 x lO -10 


-1.02421 x 10~ 9 


u 3 


-1.98609 x 10" 1 


-1.05095 x lO" 2 


-8.77204 x 10~ 4 



Figure 2.5 shows the undeformed and deformed shapes of the sandwich plate using 
40 sandwich elements. 
2.4.2. Sandwich Plate with Ply Drop-offs 

In this section, we present the computational results for sandwich plates that have 
discontinuities due to disparities in the length of the layers, resulting in the so-called ply 
drop-offs. 

2.4.2.1. Sandwich plate with ply drop-off 

We now consider a cantilever sandwich plate with three layers, and with a ply drop- 
off in the top layer at mid length, see Figure 2.6. The free edge at the tip is subjected to 



50 




Figure 2.5. Roll-down of a sandwich plate with three identical layers: Isometric view of 
deformed shape. 

a uniformly distributed force of n* 13 = 100. The geometric and material properties are 
listed below: 

L = 10, W = 0.1, T b = 0.3, T a = 0.2, (2.165) 

where L is the length, W the width, T b the thickness before the ply drop-off, and T a the 
thickness after the ply drop-off, and 

w £=1.2xl0 7 , (<)f=0., (£) k s = 0.75, for £ = -1,0,1. (2.166) 

Before the ply drop-off, the layer thicknesses are 

{e) h=0.l, for £ = -1,0,1. (2.167) 

After the ply drop-off, the layer thicknesses are 

( _D/i = 0.1, {0) h = 0.1, {l) h = 0. (2.168) 

Since the plate has a large aspect ratio, we use the Euler-Bernoulli beam theory to 
predict the deflection. The bending stiffness coefficients of the beam before and after the 



51 



A 
A 
A 

/ 



B 



L/2 



L a - L/2 



Figure 2.6. Sandwich plate with one ply drop-off. 

ply drop-off are 

£/ b = £W(T 6 ) 3 /12 = 2700, J5/ a = LW(T a ) 3 /12 = 800, 



(2.169) 



respectively. 

Let P = n* 13 W and M = PL a be the resultant tip load, and the internal moment 
at the ply drop-off. Let u\ be the transverse displacement at the ply drop-off B due to the 
force P and the moment M. Let u\ be the transverse displacement at C of thin half of the 
plate, with the section B at the ply drop-off clamped. The total transverse displacement u 3 
at the plate tip is the sum of u\ , the transverse displacement at C due to the rotation B b of 
the section at B (this rotation results from the bending of the portion AB), and u\ : 

u 3 = ul + L a 6 b + u\ 

= {PL b 3 /3EI b + ML b 2 /2EI b ) + 

(PL b 2 /2EI b + ML b /EI b )L a + PL 3 a /3EI a 
= 1.6010. ( 2 - 17 °) 

Table 2.2 presents the computed results using 20 uniformly distributed linear sand- 
wich shell elements and using 20 uniformly distributed equivalent linear single-layer shell 
elements, respectively. The transverse displacement u 3 obtained with sandwich shell ele- 
ments has a relative error of 0.030% compared with the analytical result based on Euler- 
Bernoulli beam theory, and a relative error of 0.033% compared with the computed result 



52 



using single-layer shell elements. 

Table 2.2. Sandwich plate with ply drop-off: Tip displacements 



Disp. 


sandwich elements 


singer-layer elements 


v> 


-1.71076 x lO -1 


-1.62236 x lO" 1 


u 2 


-1.43540 x lO" 13 


3.58955 x 10~ 13 


u:'' 


1.55253 


1.54842 



Remark 2.9. Similar to the case of sandwich beams in Vu-Quoc and Deng [1995], 
the result with single-layer shell elements is smaller than that with sandwich shell ele- 
ments, since the equivalent single-layer plate has a symmetric ply drop-off, unlike the 
non-symmetric ply drop-off in sandwich plate. Further, hinge is not allowed to form in 
the cross section at the ply drop-off in the equivalent single-layer shell. Figure 2.7 depicts 
the undeformed shape and the deformed shape; the effect of the ply drop-off is not easily 
discernible. I 




Figure 2.7. Sandwich plate with ply drop-off subjected to tip force: Isometric view of 
undeformed and deformed shapes. 



4 For moderate thick plate and Poisson's ratio v = 0, Euler-Bernoulli beam theory gives accurate results 
on displacements. 



53 

2.4.2.2. Two-layer plate with ply drop-off: aspect ratio A = 5 : 1 : (1, 0.5) 

Here the aspect ratio of the two-layer plate with ply drop-off is represented by A :— 
L : W : (7},, T a ). We now consider a cantilever sandwich plate with only two outer 
layers (and with the core layer inexistent), subjected to the uniformly distributed force of 
n* 13 — 60 assigned at the free edge. The plate has a ply drop-off at mid length. The 
geometric dimensions of the plate are (Figure 2.8) 

L = 5, W = l, 7; = 1, T a = 0.5, (2.171) 

with the layer thickness before the ply drop-off being 

( _i)fc=0.5, ( o)^=0.0, (1) /*= 0.5, (2.172) 

and after the ply drop-off 

(_!)/» =0.5, (0) /i=0.0, (1) h=0.0. (2.173) 
The material properties chosen are 

(/)£?= 29,000, {l) u = 0.294, (0 « s = 1, for £=-1,0,1. (2.174) 

Ten uniformly distributed elements are used in the computation. The displacements 
of a corner node at the tip are tabulated in Table 2.3. The undeformed and the deformed 
shapes are shown in Figure 2.9, where a change in curvature at the ply drop-off is clearly 
discernible. 

Table 2.3. Two-layer plate with ply drop-off, A = 5 : 1 ; (1, 0.5): Tip displacements. 



u 1 


u 2 


u 3 


-5.18243 x 10- 1 


-4.22720 x 10- 4 


1.55017 



2.4.2.3. Two-layer plate with ply drop-off: aspect ratio A = 20 : 1 : (1, 0.5) 

We now consider a cantilever sandwich plate with only two outer layers (and with the 
core layer inexistent), subjected to a tip moment. The plate has a ply drop-off at mid length 



54 




L 



Figure 2.8. Two-layer plate with ply drop-off: Geometry and assigned force. 




Figure 2.9. Two-layer plate with ply drop-off. Aspect ratio A == 5 : 1 : (1, 0.5): Isometric 
view of undeformed and deformed shapes. 

(see Figure 2.10). Since the connection between the thinner part L a and the thicker part 
L b of the plate is flexible, the actual moment needed to bend the thinner part L a into a full 
circle is a little smaller than the theoretical result of M = 2irEI a /L a , which is obtained for 
an equivalent beam with a clamped end. 

The geometric dimensions of the plate are as follows 



L = 20, W = l, L b = 10, L a = 10, T 6 = l, T a = 0.5. 



(2.175) 



The layer thickness before the ply drop-off are 



= 0.5, ( )/i = 0.0, (i)h = 0.5 



(2.176) 



55 



and after the ply drop-off 



=0.5, (0) A = 0.0, {l) h=Q.O 



(2.177) 



The material properties chosen are 

{t) E = 29000, {t) u = 0.294, {1) k s = 1. for £ = -1, 0, 1. 



(2.178) 



Along the free edge at the tip of the cantilever plate, we assign a uniformly distributed 
resultant couple (-i)m* 12 = 189.8, which corresponds to the theoretical value of the tip 
moment to bend a beam equivalent to the thinner part L a of the plate into a full circle. 



Figure 2.10. Two-layer plate with ply drop-off, Aspect ratio A = 20 : 1 : (1, 0.5): Geome- 
try and assigned couple. 

As mentioned above, since the connection between the thickness part and the thin- 
ner part at the ply drop-off of the plate is flexible, the assigned resultant couple is higher 
than what is needed to roll the thinner part of the plate into a full circle. Thus it is ex- 
pected that the tip of the plate will be rolled past the ply drop-off location. The computed 
displacements of the nodes at the ply drop-off and at the tip of the plate are reported in 
Table 2.4. 

In the case where a full circle is obtained, point B in Figure 2.10 should come back 
to coincide with point A (in a projection onto the plane (f 1 , £ 3 )); in such case, the displace- 
ment of point B should be u\B) = -10 - v}(A) = -10 - 0.969 = -10.969, where the 
value of u 1 (A) = -0.969 comes from Table 2.4. The computed displacements u 1 for point 
B is, however, equal to (-10.789), thus corresponds to a relative error of 1.6% compared 




56 



Table 2.4. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5). Displace- 
ments at the ply drop-off (point A in Figure 2.10) and at the tip (point B in Figure 2.10). 



Node 


e 


£ 2 


e 


u 1 


u 2 


u 3 


101 


10. 


0. 


0. 


-0.9689 


7.875 x 10~ 3 


-3.705 


105 


10. 


1. 


0. 


-0.9689 


-7.875 x 10- 3 


-3.705 


201 


20. 


0. 


0. 


-10.789 


8.768 x 10- 3 


-3.937 


205 


20. 


1. 


0. 


-10.789 


-8.768 x 10" 3 


-3.937 




Figure 2.11. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll 
down maneuver . Isometric view of deformed shapes (Peeling of a banana). 

to the value of (—10.969) mentioned above. The deformed shape of the plate, shown in 
Figure 2.1 1, evokes the action of peeling a banana. 

To display the effects of anticlastic curvature that the sandwich shell elements can 
capture, we refine the discretization to 160 sandwich shell elements. The three-dimensional 
rendering of the deformed shapes are given in Figures 2.13, Figures 2.14, and Figures 2.15. 
The anticlastic curvature can be seen clearly in Figure 2.13 and Figure 2.14. Since the 
top surface of the plate is stretched in the f 1 direction, when the plate is roll down, by the 
effect of the Poisson's ratio, this top surface experiences a contraction in the £ 2 direction. 
The reverse is for the bottom surface of the plate (i.e., a contraction in the direction 
and a stretching (expansion) in the £ 2 direction). The combined effect of stretching and 



57 




Figure 2.12. Ply drop-off problem. Cantilever sandwich shell with drop-off subjected to tip 
moment: Peeling of a banana. 

contracting of the top and bottom surfaces of the plate in the £ 2 direction is the result 
of bending in the f 1 direction. To quantify the anticlastic curvature, and to compare the 
result with a calculation employing 3-D solid elements using the nonlinear finite element 
code ABAQUS, we look at the difference in the transverse displacement u z of two points 
located at f 1 = 13.5 (see Figure 2.15), one point at £ 2 = (i.e., at the outer lateral edge of 
the plate), and the other point at £ 2 = 0.5 (i.e., in the middle of the plate in £ 2 direction): 

u 3 (13.5, 0.5) - u 3 (13.5, 0) = -6.68096 - (-6.70291) = 0.022 . (2.179) 

The quantity in (2. 1 79) is to be compared to the quantity in (2. 1 80) obtained from ABAQUS. 

We note that the resultant couple needed to roll the thin part of the plate into a full 
circle is (-i)m* 12 « 177.5 which is 94% of the magnitude of the tip moment needed to roll 
an equivalent clamped beam into a full circle. This lower magnitude is due to the flexibility 
by the plate at the ply drop-off line, as discussed earlier. 

To compare the results obtained with our sandwich shell element, we solve the same 
problem using the solid elements in the nonlinear finite element code ABAQUS. In our 



58 




z 

It— >a 

Figure 2.14. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): 
Roll down maneuver. 3-D rendering of the deformed configuration. Anticlastic curvature 
viewed from f 1 direction. 

ABAQUS model, we employ 960 C3D8I (8-nodes) linear brick elements, with 1453 
nodes. These elements belong to the class of incompatible mode formulation. The final 
value of the resultant couple assigned along the force edge at the tip of the plate is 189.8 



59 




= 13.5 



Figure 2.15. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): 
Roll down maneuver. 3-D rendering of the deformed configuration. Anticlastic curvature 
viewed from £ 2 direction. 

(See Figure 2.19), which corresponds to that obtained from beam theory. Figure 2.16-2.18 
provide various views of the final deformed configuration of the ABAQUS model. 




Figure 2.16. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll 
down maneuver. ABAQUS model using 960 solid elements with incompatible modes. 
Undeformed and deformed configuration. Observer's viewpoint: (1,-1,1). 



To quantify the anticlastic curvature, and to compare this quantification to the re- 
sult obtained using the sandwich shell elements, we again consider the nodes having the 
coordinates f 1 = 13.5, £ 3 = 0, which lie on the top surface of the thinner part of the 



60 



Figure 2.17. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll 
down maneuver. ABAQUS model deformed configuration. Observer's viewpoint: (1, 0,0). 



Figure 2.18. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1,0.5): Roll 
down maneuver. ABAQUS model. Undeformed and deformed configuration. Observer's 
viewpoint: (0,-1,0). 

two-layer plate. In the deformed configuration, these nodes are close to the points having 
the lowest spatial coordinate x 3 (or the z coordinate in Figure 2.15 and Figure 2.18). The 
displacements of these nodes are given in Table 2.5. 

Parallel to (2. 179) for sandwich shell elements, the anticlastic curvature in the ABAQUS 
model can be quantified using the results in Table 2.5 as follows 

u 3 (13.5, 0.5) - u 3 (13.5, 0) = -6.5948 - (-6.6159) = 0.022. (2.180) 

The above result agrees well with that obtained from the sandwich shell element. 



61 



Deformed plate 




M = F x d 



Figure 2 19 Two-layer plate with ply drop-off. Aspect ratio A » 20 : 1 : (1,0.5): 
ABAQUS solid model. Assigned forces at plate tip to create a resultant couple in the 

roll-down maneuver. . 
Table 2 5. Two-layer plate with ply drop-off. Aspect ratio A -20: 1: (1, 0.5): Anticlastic 
curvature from ABAQUS solid model. Nodal displacements of nodes having coordinates 
f 1 = 13.5, £ 3 = 0. 



e 


e 


u 1 


u 3 


13.5 


0.00 


-6.007 


-6.780 


13.5 


0.25 


-6.007 


-6.764 


13.5 


0.50 


-6.007 


-6.758 


13.5 


0.75 


-6.007 


-6.674 


13.5 


1.00 


-6.007 


-6.780 



For the thinner part of the two-layer plate to roll into a complete circle, point B in 
Figure 2.10 must roll back to coincide with point A which is itself moved by the deformed 
plate. To compare the results obtained using sandwich elements and those obtained using 
an ABAQUS model, we gather the coordinates of point A and B in the final deformed 
configuration, corresponding to the resultant couple of M = 189.8 in Table 2.6 below. 
2.4.2.4. Tw o-layer plate with ply drop-off: aspect ratio A = 20 : 10 : (1, 0-5) 

We now consider the same plate as in the previous section, but with a width ten 
times larger (i.e., W = 10.), instead of W = 1. as in the previous section. All other 
geometric dimensions and material properties remain identical to those in the previous 



62 



Table 2.6. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 1 : (1, 0.5). Compari- 
son between sandwich elements and ABAQUS solid model. Distance between point A and 
point B in the final deformed configuration. £ A = (10, 0, 0) , £ B = (20, 0, 0) 











Sandwich elements 


9.03 


0.00788 


-3.71 


ABAQUS model 


8.64 


0.0167 


-3.56 








<F(€a) 


* 3 (£b) 


Sandwich elements 


9.21 


0.00877 


-3.94 


ABAQUS results 


8.61 


0.00930 


-3.53 



Distance between A and B = * ( £ /i) ~ * ( £ b)| 


Sandwich elements 


0.293 


ABAQUS results 


0.0459 



section, see Eqs. (2.175), (2.176), (2.177), and (2.178). The distributed couple ( -i)m* 12 
assigned to the free edge of the plate tip is set as before to (-i)m* 12 = 189.8, which is 
the resultant couple that will roll an equivalent beam into a full circle. To discretize the 
two-layer plate, we now employ 200 sandwich shell elements: 100 elements before the ply 
drop-off, and 100 elements after the ply drop-off. The deformed shapes of the plate are 
shown in Figure 2.20- 2.23. Figure 2.20 depicts the deformed shape in an isometric view. 
One can clearly see the anticlastic curvature in the £ 2 direction in this figure, as well as 
in Figure 2.22, which is the projection of the deformed shape on the (£ 2 , £ 3 ) plane. This 
anticlastic curvature is the effect of the Poisson's ratio. The top surface of the undeformed 
plate is extended in the £* direction, this extension induces a contraction in the transverse 
£ 2 direction, and thus the downward curvature is clearly seen at the bottom of the deformed 
plate in Figure 2.22. Opposite to what take place at the top surface, the compression of the 
bottom surface of the plate in the £ x direction induces an extension in the £ 2 direction, thus 
resulting in the lateral bulging of the plate, as seen in Figure 2.22. 

To quantify the anticlastic curvature similar to (2.179) and (2.180), we consider the 
transverse displacement u 3 of the nodes at £ r = 14 and £ 3 = 0, which lie near the bottom 




z 

III ,x 

Figure 2.20. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1, 0.5): Roll- 
down maneuver. Sandwich shell elements 3-D rendering of deformed shape. Isometric 
view. 



B 




A 

Figure 2.21. Two-layer plate with ply drop-off. Aspect ratio A — 20 : 10 : (1,0.5): 
Roll-down maneuver. Zoom-in on the ply drop-off point. 

of the deformed configuration of the two-layer plate (see Figure 2.23) 



u 3 (U, 5) - u 3 (14, 0) = -6.62581 - (-6.64764) = 0.02183. (2.181) 



z 



Figure 2.22. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1,0.5): 
Roll-down maneuver. Sandwich shell elements. 3-D rendering of deformed configuration. 
Projection down the f 1 axis. 

To clearly depict the deformation at the ply drop-off and the tip of the deformed plate 
around the area of the ply drop-off, we provide a zoom-in figure on this area in Figure 2.21. 
The distance between point A and point B will be used to compare the results obtained 
with sandwich shell elements and those obtained with an ABAQUS solid model, which 
is composed of 2400 incompatible (solid) linear brick elements (type C3D8I). The final 
moment at the plate tip assigned to the ABAQUS solid model has a magnitude of 1898, 
which is obtained for the roll-down of an equivalent beam. Various views of the deformed 
configuration obtained with the ABAQUS model are depicted in Figure 2.24, Figure 2.25, 
Figure 2.26. The distance between point A and point B in the deformed configuration as 
obtained both from sandwich shell elements and from the ABAQUS model are given in 
Table 2.7. 

From the above examples, we found that the present sandwich shell formulation gives 
good results on displacements with very coarse mesh, when compared to the 3-D converged 
results from ABAQUS. For the interlaminar stress analysis, due to the kinematic assump- 



65 




Figure 2.23. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1,0.5): 
Roll-down maneuver. Sandwich shell elements. 3-D rendering of deformed configuration. 
Projection down the £ 2 axis. 




Figure 2.24. Two-layer plate with ply drop-off. Aspect ratio A — 20 : 10 : (1,0.5): 
Roll-down maneuver. ABAQUS solid model. Undeformed and deformed configurations. 
Isometric viewpoint : (-4,-7,-3) 



tion, the present formulation is expected to be more accurate than the single-layer shell 
model, especially for thick and moderate thick shells. 



66 




Figure 2.25. Two-layer plate with ply drop-off. Aspect ratio A - 20 : 10 : (1,0.5): 
Roll-down maneuver. ABAQUS solid model. Projection down the f 1 axis of deformed 
configuration. 




Figure 2.26. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1, 0.5): Roll- 
down maneuver. ABAQUS solid model. Projection of the deformed configuration along 
the £ 2 axis. 



67 



Table 2.7. Two-layer plate with ply drop-off. Aspect ratio A = 20 : 10 : (1, 0.5). Compari- 
son between sandwich elements and ABAQUS solid model. Distance between point A and 
point B in the final deformed configuration. £ A = (10, 0, 0) , £ b = (20, 0, 0) 









$ 3 Ua) 


Sandwich elements 


9.16 


0.0112 


-3.5 


ABAQUS model 


9.16 


0.0179 


-3.53 












Sandwich elements 


9.07 


-0.0345 


-3.41 


ABAQUS results 


8.23 


-0.0294 


-2.83 



Distance between A and B = $ ( f a) ~ * ( £ b)\\ 


Sandwich elements 


0.142 


ABAQUS results 


1.17 



CHAPTER 3 

OPTIMAL SOLID SHELLS FOR NONLINEAR ANALYSES OF 
MULTILAYER COMPOSITES : STATICS 

3.1. Introduction 

The analysis of general shell structures have been of interest for several decades. 
There is a continuing need to develop more reliable, accurate and efficient shell element, 
especially for analyses of composite structures covering a wide range of physical scales (in- 
cluding MEMS 5 ) and material and geometric nonlinearities. Structures made of laminated 
composites continue to be of great interest for engineering applications. For accurate anal- 
yses of composites with a large number of layers, industry routinely employs FE meshes 
with one solid element per ply in the thickness direction, and with element aspect ratio less 
than 10 (Figure 3.1). It is therefore highly desirable to develop efficient finite elements that 
are accurate at extreme aspect ratio to significantly reduce the computational effort. 




Figure 3.1. Composite structure with 500 plies in the thickness direction; the ply thickness 
is around 10 -3 zn. 

Shell element formulations have been mainly developed within the context of the 
so-called degenerated shell concept and the classical shell theory (Buechter and Ramm 
[1992]). Both formulations are based on the common kinematic assumptions of inextensi- 
5 MEMS stands for MicroElectroMechanical Systems. 

68 



69 



bility in the thickness direction and the zero transverse normal-stress condition. 6 Although 
these approximations led to very good results in most cases, several difficulties could arise: 
(i) Complex 3-D material models: the zero transverse normal stress condition must be im- 
posed, (ii) Boundary conditions and finite rotations: use of rotational degrees of freedom 
(dofs); especially those normal to the boundary, to describe soft support and hard support 
(e.g., Zienkiewicz and Taylor [1991, p.92]); complex update algorithms for finite rota- 
tions in geometrically-exact stress-resultant formulation (e.g., Vu-Quoc and Deng [1995], 
Vu-Quoc and Ebcioglu [1996], Vu-Quoc and Ebcioglu [2000a], Vu-Quoc and Ebcioglu 
[20006], Vu-Quoc, Deng and Tan [2000]). (iii) Transverse normal stress: inconsistently a 
posterior computation based on the computed in-plane stresses (see Reddy [1997, p.345], 
and e.g. in the localized effects due to the concentrated surface loading and the delam- 
ination of composite shells), (iv) Combination with regular solid elements: Transition 
elements are needed to connect rotational dofs and displacement dofs (e.g., Kim, Varadan 
and Varadan [1997] and the contact problem), (v) Through-the-thickness stress distribu- 
tion in laminated composite with dissimilar materials: poor accuracy because of straight 
director assumption (Bischoff and Ramm [2000]). 

Accurate and robust low-order shell elements have always been in high demand for 
development and for use in engineering analysis (e.g., DYNA3D [1993], NIKE3D [1995]), 
particularly when complex nonlinear 3-D constitutive relations can be incorporated with- 
out the added requirement to satisfy the constraint of zero transverse normal stress. Three 
possible shell kinematic descriptions have been proposed: (i) The displacement of the refer- 
ence surface together with the extensible transverse director (Simo, Rifai and Fox [1992], 
Betsch, Gruttmann and Stein [1996]). (ii) The displacement of the reference surface to- 
gether with the displacement vector of the tip of a director (Braun, Bischoff and Ramm 
[1994], Roehl and Ramm [1996], Bischoff and Ramm [1997], Bischoff and Ramm [2000]). 
(iii) The displacement of the top and bottom surface of the shell (e.g. Hauptmann and 

6 Stress-resultant shell formulation can be generalized to account for thickness change, which relaxes 
the zero transverse normal stress condition (e.g., Simo, Rifai and Fox [1992]). 



70 



Schweizerhof [1998], Klinkel, Gruttmann and Wagner [1999], Ramm [2000]). The kine- 
matic descriptions (ii) and (iii) are attractive since they avoid the complex rotation updates 
in stress-resultant elements. On the other hand, the kinematic description (iii) provides a 
natural way to connect to regular solid elements without the need for transition elements; 
such feature can benefit the detailed modeling of shells with patches of piezoelectric or 
viscoelastic materials. 

The present solid-shell element has the same displacement dofs as in regular lin- 
ear (8-node) brick solid element. Displacement-based solid elements are known to have 
poor performance in bending-dominated situation, such as in thin shells. To obtain the 
same performance as stress-resultant shell formulations with plane stress assumption (e.g., 
Vu-Quoc, Deng and Tan [2000]), the Enhanced Assumed Strain (EAS) method and the 
Assumed Natural Strain (ANS) method are employed here. 

To improve the bending behavior of low-order elements, the EAS method based on 

the Fraeijs de Veubeke-Hu-Washizu functional was proposed by Simo and Rifai [1990]. For 

large deformation analyses, there are two ways to introduce the EAS method: (i) enhancing 

the deformation gradient F (Simo and Armero [1992], Miehe [1998b]), and (ii) enhancing 

the Green-Lagrangian strain tensor E (Bischoff and Ramm [1997], Klinkel and Wagner 

[1997], Klinkel et al. [1999] etc.). From the computational standpoint, the latter is simpler 

and more efficient, even though our numerical experience indicates that both approaches 

lead to the same numerical results when the same EAS parameters are used. 7 To incorporate 

3-D constitutive laws in shell formulations, the transverse normal strain must have at least 

a linear distribution over the shell thickness; otherwise, the so-called Poisson-thickness 

locking would occur (Zienkiewicz and Taylor [1991, p. 161], Bischoff and Ramm [1997]). 

To relieve the Poisson-thickness locking, two methods were proposed in recent years: (i) 

Assuming a quadratically distributed displacement field over the shell thickness (Parisch 

[1995]), which then introduced an additional kinematic parameter, and (ii) using the EAS 

7 Noted that the EAS method based on the displacement gradient as proposed in Miehe [19986] and used 
in Miehe and Schroeder [2001] does not pass the bending patch test, and there is no easy way to remedy this 
problem, see the next paragraph for the details. 



71 



method to enhance the transverse normal strain (Buchter, Ramm and Roehl [1994]). In 
our formulation, we enhance the transverse normal strain by the HAS method to include 
bilinear terms ^ 3 and £ 2 £ 3 in terms of material coordinates. To improve the membrane 
bending behavior, we also enhance the membrane strains in the similar manner as in Simo 
and Rifai [1990]. 

On the other hand, to make the formulation more efficient, we propose a modified 
HAS method while still keeping the same level of accuracy, where the inverse of element 
Jacobian matrix and the Jacobian at the element center are no longer necessary. Further- 
more, the present eight-node solid-shell element relies on a new optimal seven-parameter 
EAS-expansion (for the transverse normal strain and for the membrane strains) together 
with an ANS method (for the transverse shear strains); the present formulation is shown 
to pass both the membrane patch test and the out-of-plane bending patch test. It should be 
noted, however, that while the 30-parameter HAS expansion of Klinkel and Wagner [1997], 
the five-parameter HAS expansion of Miehe [1998*] and of Klinkel et al. [1999] pass the 
membrane patch test, all of them fail to pass the important out-of-plane bending patch test. 

For the HAS approach using enhancing deformation gradient, we develop an HAS 
expansion by superposing the enhancing converted basis to the compatible converted basis, 
and then present a formulation that is much simpler than that employed in Miehe [1998*] 
(see Section 4.3). 

Two ANS modifications on the compatible covariant strains are employed to elimi- 
nate the locking effects from the compatible low-order interpolations. ANS interpolation is 
the most successful tool to overcome the shear-locking effect in the 4-node displacement- 
based shell elements, even for initially distorted meshes (MacNeal [1978], Hughes and 
Tezduyar [1981], Dvorkin and Bathe [1984], Parisch [1995]). We apply an ANS inter- 
polation of the compatible transverse shear strains to treat shear locking. In the case of 
curved structure with geometric nonlinearity, there is another locking effect: The so-called 
curvature-thickness locking (Bischoff and Ramm [1997]), which is also known as the trape- 
zoidal locking (Sze and Yao [2000]); this type of locking can be avoided by introducing the 



72 



ANS interpolation of the compatible transverse normal strain, as proposed by Betsch and 
Stein [1995]. Such treatment can improve the performance of the formulation in Parisch 
[1995] and Hauptmann and Schweizerhof [1998]. 

The features of this solid-shell formulation are summarized as below: 

. The kinematic description involves only displacement dofs that require no complex 
finite rotation update and no transition elements to connect solid-shell elements to 
regular solid elements (Hauptmann and Schweizerhof [1998]). 

. The use of covariant Green-Lagrange strain tensor, without neglecting any higher 
order terms (e.g., as in Bischoff and Ramm [1997]). The stress and strain terms 
quadratic in £ 3 become important in the analysis of relatively thick shells, for strong 
curvatures or in the presence of large strains together with bending deformations 
(Buchter et al. [1994]). 

. All stress and strain components are accounted for, thus allowing for an implementa- 
tion of unmodified 3-D nonlinear constitutive laws, without the need for applying the 
plane-stress constrain. The strain-driven character of the formulation also makes it 
easier to implement nonlinear constitutive models, as compared to the hybrid finite- 
element formulations (Simo, Kennedy and Taylor [1989]). 

. In contrast to EAS formulation based on the deformation gradient F (see Section 4.3), 
EAS formulation based on enhancing the Green-Lagrange strain tensor (together 
with the use of the second Piola-Kirchhoff stress tensor) are much simpler (Simo 
and Armero [1992] and Andelfinger and Ramm [1993]). 

. An ANS method applied on the transverse shear strains is used to relieve the trans- 
verse shear-locking problem (Dvorkin and Bathe [1984]), whereas an ANS method 
applied on the normal strain components is used to remedy the curvature thickness 
locking problem (Betsch and Stein [1995]). 



73 



In addition to the above features, our new contributions to the field are specifically listed as 
below: 

• Optimal (minimum) number of EAS parameter to pass the patch tests for both the 
membrane response, and the out-of-plane bending: (i) three EAS parameters on the 
transverse normal strain to remedy the Poisson-thickness locking, and (ii) four EAS 
parameters on the membrane strains to remedy the in-plane bending behavior. 

• Efficient EAS method that avoids the computation of the Jacobian at the element 
center, and that no inverse of the Jacobian matrix at the element center is needed. 

• By using the tensor form, we prove the equivalence of the 2-D plane elasticity el- 
ements of Simo and Rifai [1990], of Taylor, Beresford and Wilson [1976], and our 
new enhancing formulation. 

• We justify through numerical experiments the relative importance of the separate 
use of the EAS method and the ANS method, as compared to the pure displacement 
formulation, and more importantly, the combined use of both the EAS method and 
the ANS method in obtaining accurate results for plate bending problem over a large 
range of aspect ratios. 

The comparison among the above various solid-shell formulations is listed in Table 3.1. 8 



Table 3.1. Comparison of various solid-shell concepts. 





Bending 
Patch test 


Loc king-free 


Absence of rot. dofs/ 
Disp. dofs only 


Higher-order terms 
in thickness coord. 


Absence of 
pre- integration 


Model parameter- space 
dimension 


Optimal 
EAS 


Present clement 


yes 


yes 


yes 


yes 


yes 


3-D 


yes 


RammcL a].[1997) 


yes 


yes 


yes 


no 


no 


2-D 


no 


Schweizcrhof el. al.|1998] 


yes 


no 


yes 


yes 


yes 


3-D 


no 


BelscheL al.|!996] 


yes 


yes 


no 


yes 


yes 


2-D 


no 


Miehe |1998] 


no 


yes 


yes 


yes 


no 


3-D 


no 



Parameter-space dimension is defined as: 1-D (beam), 2-D (stress-resultant plates and shells) and 3-D 
(solids); Deformation-space dimension is defined as: 2-D (planar deformation), 3-D (general deformation) 



74 

The outline of the present chapter is as follows. After a presentation of the kinematics 
assumption and the formulation based on the Fraeijs de Veubeke-Hu-Washizu (FHW) vari- 
ational principle (Felippa [2000]) in Section 3.2, we discuss the finite-element discretiza- 
tion and its implementation in Section 3.3. A review of the EAS method together with 
our proposed modification is presented in Section 3.4. We present the numerical results in 
Section 3.5. 

3.2. Kinematic Assumption and FHW Variational Formulation 
The extension of the EAS method to geometrically non-linear problems by Simo and 
Armero [1992] employed an enhancement of the deformation gradient F and thus a multi- 
plicative decomposition An alternative line of formulation for geometric and material non- 
linearities based on the enhancement of the Green-Lagrange strain E leads to particularly 
efficient computational effort (see, e.g. Bischoff and Ramm [1997]), with practically the 
same results. We describe below the kinematics of a solid shell in curvilinear coordinates 
and review the three field FHW variational principle and its role in the EAS method. 
3.2.1. Kinematics of Solid-Shell in Curvilinear Coordinates 

To overcome the known problems associated with the rotational degrees of freedom 
in traditional shell elements, the shell kinematics of deformation is described by using the 
position vectors of a pair of material points at the top and at the bottom of the shell sur- 
face. In this kinematic description, a straight transverse fiber before deformation remains 
straight after deformation. Such transverse fiber between two corresponding nodes at top 
and bottom surfaces needs not be normal to the shell mid-surface before deformation, as 
well as after deformation. We distinguish here three configurations of the shell: (i) the 
material configuration, which is the biunit cube, (ii) the initial (or undeformed) configura- 
tion, which could be curved, and (iii) the current (or deformed) configuration (see Fig.5 in 
Vu-Quoc and Ebcioglu [20006]). The initial (undeformed) three-dimensional continuum 
of the shell geometry (Figure 3.2) is described by 



75 

□:«H,l]xH,l}xH,U, (3.1) 

where X (•) is the mapping from the biunit cube □, parameterized by the material coordi- 
nation (f 1 , £ 2 , £ 3 ), to the initial configuration. The image X (£) of a point £ = (f 1 , £ 2 , £ 3 ) 6 
□ is next represented by a linear combination of the position vectors X u (•, •) and Xi (•, •) 
of a point in the upper material surface £ 3 = + 1, and a point in the lower material surface 




Figure 3.2. Initial (undeformed) configuration of solid shell: convective coordinates f and 
position vectors X u and X/. 

The kinematics of the present formulation is the same as in (Hauptmann and Schweiz- 
erhof [1998]), and related to the director formulation (e.g. Bischoff and Ramm [1997]), if 
one rewrites (3.1) as follows 

xw = I [x.{e,e)+x, (e,e)} +le[x.{t 1 ,?)-x, (?,?)} 

where X m is the position vector of the mid-surface in the initial configuration, D the unit 
director, and h (£*, £ 2 ) the shell thickness. 



76 



Similarly, in the current (deformed) configuration, the geometry of the solid shell is 



described by 

1 
2 
1 

2 



;«.(^)+*i(^,c a )]+-5? m*\ (^,e 2 ); 
;(i+«*)«-(€ 4 .^)+(i-^)*»(^) 



(3.3) 



where ar(-) is the mapping from the biunit cube parameterized by (£\£ 2 ,£ 3 ) (material 
configuration), to the current (deformed) configuration, x u (•, •) andxj (•, •) the position 
vectors of the deformed upper and lower surface of the solid shell, respectively. 

The initial configuration is related to the deformed configuration (Figure 3.3) by the 
displacement field u as follows 

x(t) = X(£) + u($). (3.4) 

The convected basis vectors G h i = 1, 2, 3, in the initial configuration B are related 
to the position vector X and the converted coordinates <f by 

dx(0 



and satisfy the following relations 



2 = 1,2,3, 



(3.5) 



Gi'G j = S{ , Gi • Gj = dj , i,j = 1,2,3, (3.6) 

where Gy are the components of the metric tensor with respect to the basis G { <g> G j 
in the initial configuration B . configuration. To simplify the presentation, we will omit 
the argument (£) and simply write G { and G\ The covectors G i can be obtained by the 
following relation 



G l = G j Gj , with 



Qij 



= [Gij]- 1 € 



(3.7) 



similarly, the convected basis vectors gi in the current configuration B t are obtained by 
using (3.4) and (3.5) as follows 



dx du 



(3.8) 



77 




B 

Figure 3.3. Solid-shell: material configuration B, initial configuration Bo, and deformed 
configuration B t . 

and satisfy the following relations 



9 % -9 1 = % , g.-Qj = 9ij , i, j = 1, 2, 3 , 



(3.9) 



where c/jj are the components of the metric tensor with respect to the basis g { <g> g j in the 
current configuration B t . The covectors g i can be obtained as follows 



g l = g l3 gj , with [<^j = [^-p e 



-1 ^ hd3x3 



(3.10) 



The deformation gradient expressed in converted basis vectors g { and takes the 



form 



(3.11) 



Using (3.1 1), (3.6), and (3.9), we can then write the (compatible) Green-Lagrange strain 



78 

tensor with respect to the convective coordinates as follows 

E c = l - [F T F - I 2 ) = i [(G* ® g) ( 9j G j ) - CjjG' ® G j 

= \ im ~ G l ®G j = E^G { <g> G J , (3.12) 

where is the covariant components of the strain tensor E c , and JT 2 the identity tensor 
expressed with respect to the convected basis G l ® G-* using the relations e p — Gf G ! and 
e 9 = GjG- 7 as follows 

ff 2 = <$ P<? e p ® e 9 - ^G^GjG* ® G J = GJgJG* ® G J = GjjG' G J , (3.13) 

Using (3.8), the metric-tensor components in (3.9) 2 can be expressed in terms of 
the convected basis vector Gj and the displacement vector u as follows 



_ _ du du _ <9w <9u 

Substituting (3.14) into (3.12), the covariant components 25& of the compatible Green- 
Lagrange strain tensor E c read as 

The second Piola-Kirchhoff stress tensor S is conjugated with the Green-Lagrange 
Strain E c , and can be expressed with respect to the basis G { <g> Gj as follows 

S = S^G l ®G j , (3.16) 

with 5 U being its contravariant components. 
3.2.2. Variational Formulation of EAS Method 

In this section, we provide a brief overview of the EAS method, which has the three- 
field Fraeijs de Veubeke-Hu-Washizu (FHW) 9 variational principle with the following 

9 We refer the readers to Felippa [2000] for a history on the contribution of Fraeijs de Veubeke to the 
formulation of multifield variational principles. 



79 



functional as the point of departure 




«• (b* - u) pdV + / (u* - u) -idS - / wt*dS 




(3.17) 



where the displacement it, the Green-Lagrange strain and the second Piola-Kirchhoff 
stress S are the independent variables. In (3.17), the expression of E c (u) in terms of 
u is given in (3.12) and (3.15); W s is the stored strain energy, t the traction vector, u* 
and t* the prescribed displacement on the boundary S u and the prescribed traction on the 
boundary S a , respectively. All variables here are expressed in the initial configuration B , 
with S a l) S u = dB (i.e, the closure of the union of the boundary S u and the boundary S a 
forms the complete boundary of B ). 

The next step in the EAS method is to introduce an enhancement (or enrichment) to 
the Green-Lagrange Strain E° and, as a result, of the variation SE C as follows 



8E = 6E C + SE , 

where E and SE represent the enhanced strain and its variation, whereas E and SE 
sent the enhancing strain and its variation. Introduction of (3.18) into (3.17) yields 



E = E C + E , 



(3.18) 





(3.20) 



where it is noted that the traction t is actually a function of 5(see also Malvern [1969, 
p.69]), and not an independent variable. 



Consider a perturbation of the displacement u as follows: 



u e (X, t) = u (X, t) + eSu (X) , (3.21) 

since the variation Su is only a function of space, and not of time, the variation of the 
acceleration u is then 

Su = , (3.22) 
therefore the variation of U in (3.20) with respect to the displacement u is 



d_ 

de 



u(u S: E,s) = /^f: (^^-'5u\dV- JSu'(b*-u)pdV 

- J SwtdS - J 5wt*dS , ySu, (3.23) 
dW s 

Since the second-order tensor — — - is symmetric, the first term in (3.23) can now be rewrit- 

uE 

ten as follows 

= ^f"^")-*^)*' (3 - 24) 

where the first equality is shown in Remark 3.1, and the second equality is obtained with 
Leibniz rule. 

Remark 3.1. With the deformation gradient F (u) expressed with respect to the basis 
ej (8) Ej (see Figure 3.3), we obtain the following relations 

F m F'ja ® E J , F T = F)E J ® et , (3.25) 

F T F = FigijF'E 1 <g> E J , with g ij = e l -e j , (3.26) 
dF dF* 

SF = —8u = -^-ei®E J 8u. (3.27) 



Note that, if {ei, e 2 , e 3 } were a system of orthonormal vectors, we would have g l i = 6ij, 
however, in general, it is not necessary to have {ej, e 2 , e 3 } orthonormal. In this remark, 
we retain the notation cfi for generality. We define a symmetric second-order tensor S as 

5 := Je = sABEa ® Eb ' (128) 

By using (3.28), (3.12), and (3.26), we can then rewrite the first term in (3.23) as follows 

Since S AB = S BA from the symmetry of S, (3.29) becomes 

^:6E< = F\S AB giJ ^6u, (3.30) 
With the following component form of FS using (3.25)i and (3.28), that is, we obtain 

FS = F l jS JB ei®E B , (3.31) 
and then the following expression 

BF f) 
FS : 5F = FSl —Su = F l A S AB gij ^8u . (3.32) 
ou J du 

Comparing (3.32) to (3.30) and using SF = GRAD (Su), we arrive at 

dW 

~ : SE C = FS : SF = FSl GRAD (6u) . (3.33) 
oh/ 

I 

Using the divergence theorem , the integration of the first term in (3.24) becomes 

/ Div { f ^e} Su ) dV= I { F ^ Su ) ' ndS ' (334) 

B ^ ' dB * ' 

where n is the outward normal vector to the surface 8B . 



82 



With (3.24) and (3.34), we can write the variation of II in (3.23) with respect to u as 

|n(«.,S,s)| - / n ( F ^\. SudS -Jn 1 J F m.su < tv 

- J Su- (b* - u) pdV - J Su-tdS - J 8wt*dS 



=/ 

-i 

Bo 



n* | 




Div 





mM8+ J [-('?*)-* 

Sa ' 



'SudS 



+ pb* — p'u 



'5udV = 0, V5u, 



(3.35) 



thus the Euler-Lagrange equations resulting from the variation of IT with respect to u in 
above (3.35) are 



( dW s \ 



Div 



+ p6* = pit in £> , 



(3.36) 
(3.37) 
(3.38) 



Then make the variation of IT of (3.20) with respect to E by the perturbation 



it follows that 



E £ = E + eS E , 



(3.39) 



l n =/( 



' aw. 

dE 



- s : , v<5£ 



(3.40) 



the Euler-Lagrange equation associated with (3.40) is 



dE ' 



(3.41) 



Taking the variation of II of (3.20) with respect to S by the perturbation 

S £ = S + e6S , 



(3.42) 



83 

we obtain 

II (uE, S e ) m / 6S : EdV + j (u*- tt) -StdS , V6S , (3.43) 



6=0 Bo S u 



where it is noted that t depends on S, then the Euler-Lagrange equations associated with 
(3.43) are 



E = 0, (3.44) 
u = u* on S u , (3.45) 

where in general E when the finite element approximation is introduced. 

3.3. Finite-Element Discretization 

In this section, we present the weak form and finite-element approximation of the 
proposed solid-shell element. The orthotropic constitutive law of laminated composite is 
then derived in convective coordinates. To avoid shear locking, we use the assumed natural 
strain (ANS) method by Dvorkin and Bathe [1984] for the transverse shear strains. To 
remedy the curvature-thickness locking (Bischoff and Ramm [1997]), we adopt the ANS 
method by Betsch and Stein [1995] for the transverse normal strain. 
3.3. 1 . The Weak Form of Modified Two-Field FHW Functional 

By designing the approximation for the stress field S and the approximation for the 
enhancing strain E c such that the following orthogonality condition holds: 

/ 5 : EdV = . (3.46) 

Bo 

then the number of independent variables in the functional II in (3.20) is reduced to two, 
that is 

n (u, e) = jw s (E c («) + E) dV 

Bo 

- y u- (6* -u)pdV + J (u* -u)-t(S)dS- J u-t*dS , (3.47) 

Bo 5 U S, 



84 

leading to the following total variation 

SU («, E) = 6U mass (u) + 8U stlff (u, E) + SU ext (u) = , (3.48) 
where from (3.35) and (3.40), we have 

SU mass = Jdu-updV, (3.49) 

Bo 

5n stiff (u, E) = j (SE C (u) + 5E) : -^W s (E c (u) + E)dV , (3.50) 

Bo 

SU-txt = - j 6u-b*pdV - J 5wt*dS . (3.51) 

3.3.2. Spatial Discretization 

Let the initial configuration B be discretized into nonoverlapping nel elements B^ 

nel ,s 

with numnp nodes, such that B « |J Ba . Let /i denotes the characteristic size of the 

e=l 

finite element discretization. 

In the element domain Bq, the displacement u, its variation Su, and increment Aw 
are interpolated as follows 

u^u h = N(Z)S e \ (3.52) 
8u ^Su h = N (£) W (e) , Au w Aw' 1 = AT (£) Ad (e) , (3.53) 

where AT is a matrix containing the basis functions restricted to element Bq \ and S e) € 
R 3x numnp a matr j x containing the nodal displacements. The readers are referred to the 
Appendix A.l for the details. 

The velocities u and accelerations u are also interpolated by using the same shape 
functions and the corresponding nodal values that is 

h « & = N (0 d (e) , u^u h = N(i%) d (e) . (3.54) 

In what follows, for simplicity, we will omit the superscript h in u h , and simply write u. 



Within a typical element (e), the variation and the increment of the (compatible) 
Green-Lagrange Strain E c is related to the variation and the increment of displacement, 
respectively, based on (3.15), as follows 

fa&Li = B KO 6di£) - { A ^} 6xl - B ( die) ) ^ , (3.55) 

where the components of had been arranged into a 6 x 1 column matrix according to 
the Voigt ordering (Brillouin [1946, p.221]) 

} = {E c n , E c 22 , 2E C 12 , E c 33 , 2E C 23 , 2E C 13 } T , (3.56) 

and where B is the deformation-dependent displacement-to-strain operator, where detailed 
expression is given in the Appendix A. 1 . 

We denote the admissible variation of the element EAS parameter column matrix 
ojW g r»mw associated with the enhancing strain by 6a&, where neas is the num- 
ber of EAS parameters a (e) . Interelement continuity is not required for the enhancing 
strain E, where components can be approximated via an enhancing strain interpolation 
matrix Q and the element EAS parameters a (e) ; an interpolation applies to the variation 
and the increment of E, that is 

{4} 6X1 = <5 (0 " (e) , {SE tJ } 6xi = 9 (0 Sa^ , {A4) 6xi = g $ Aa<< (3.57) 

The number of internal parameters ot^ and the interpolation matrix g will be discussed in 
Section 3.4. 

3.3.3. Linearization of the Discrete Weak Form 

The consistent tangent operator in the Newton solution procedure is constructed by 
taking the directional derivative of the weak form at a configuration {k) u in the direction 
of the increment A {k) u, where the left subscript k designates the iteration number. The 
tangent operator can be viewed as the summation of the material and geometric tangent 
operators. The geometric part results from taking the variation of the geometry while hold- 
ing the material constant, whereas the material part results from taking the variation of the 
material while holding the geometry constant. 



86 

Applying a standard finite-element procedure to discretize the weak form (3.48), we 
obtain the following expression at the element level for the static case (u = 0) 

6U^=SU%+8U^ t = 0, (3.58) 

with the stiffness part SU^ lff from (3.50) and the external-force part SU { ^ t from (3.51) 
written as follows 

5U% = J 6{E^ T {s^}dV+ I 5 {E 13 } T {S^} dV , (3.59) 

SUii = - J 5u-b*pdV- J Su-fdS, (3.60) 

where we used the symbol definition, which corresponds to the second Piola-Kirchhoff 
stress, 

s ■= a£> (3 - 61) 

to alleviate the notation, and where the column matrix {S ij } has its coefficients arranged 
in the same Voigt ordering as in (3.56) 

{S«} m [S n ,S 22 ,S 12 ,S 33 ,S 2 \S 13 ] T . (3.62) 

It should be noted that the symbol S in (3.61) is simply used in the place of - , and is 

dE 

not the independent variable in the Euler-Lagrange equation (3.41). 

Remark 3.2. The linearization of the weak form 8U. (u, E^j can be accomplished by 
the truncated Taylor series about the kth iterate ( , ( k )E ) : 

<5n( (fc+1 )u, (k+i)E) « sn(( k )u, (k) E) 

d(8U) 



• (Au, AE) 

(u= (k) u,E= w E^j 



d (it, E) 

= <5n( (A) u, [k] E) -f Z>(<ffi)( (fc)U , {k) E) • [Au,AE) , (3.63) 



87 



where Au = {k+1) u - {k) u, AE = {k+l) E - {k) E . To compute the increment 
(Aw, AE) in the Newton's solution process, we simply set the expression in (3.63) to 



zero. 



To alleviate the notation, we will omit the left subscript k designating the iterative index. 

Using the approximation (3.52), (3.53), (3.55) and (3.57) in (3.63), the increments 
Ad [e) and Aa< e » can be computed in the Newton's solution process, as mentioned above, 
by using the following equation 

V(6IIU) (*>,««) • (AdM Aa<->) = ( Ad « Aa (e)) 
in which the variation SU% in (3.59) and SIlQ in (3.60) now take the form 

(3.65) 



(d« , tt W) 




with f% 


= J B T {s ii }dV, 


Ae) 

J EAS ~ 


aiS (<* (e) ) 


UUl J ext > 




with fi e x \ 


= f N T b*pdV+ I 


N T t*dS 



8 



:<<=> 



8. 



.00 



(3.67) 
(3.68) 



Thus the left hand side of (3.64) becomes 

V 1 K ' 8d (e) daW 

= K )T *2 + toW^fcW] , Ad (e) + [sd^kH + fa^SS] -AaW 

= [fcW AdW + fcW AaW] + feW [*W ArfW + *£> Aa M] , (3 69) 

Let the matrix of tangent elastic moduli C be defined as 



c = 


Qijkl 












dE kl \ 



j6x6 



(3.70) 



88 

where C ljkl are the components of constitutive tensor C in the convected basis, and are 
subsequently arranged in the matrix C according to the ordering of the strain components 
in (3.56) and of the stress components in (3.62). 

Using (3.69), (3.66)i, (3.70), and (3.55), we obtain the following expressions 

df (e) 



*S = = / (G r S + B T CB) dV , (3.71) 

k ™ - - 1 BTcgdV • < 3 - 72 > 



s <«) 



where the matrix 



dB (S e) ) 



(which is a function of the coordinates £), and the stress matrix $ (which is related to the 
matrix {S* j } in (3.62)) have their detailed expressions given in the Appendix A.l. It is 
noted that the dimension of $ and G are 144 x 24 for the present element with six stress 
components, eight nodes per element, and three dofs per node. From (3.69), (3.66) 2 , (3.70), 
(3.73), and (3.55), we obtain the remaining parts of the stiffness matrix 

fc (e) °J EAS \jJe)] T f r TnnA\r 't***\ 

k au - (e) - [*^J = j y CBdV , (3.74) 



df 



(e) 
EAS 



= J Q T CgdV. (3.75) 



It follows from (3.64), (3.65), (3.67), and (3.69) that the discrete linearized system 
of equations to solve for the increments Ad (e) and Aa (e) is given by 

fcWAdW + fcWAaW = f%-f%, (3.76) 

fcWAifW + fcWAaW = (3.77) 

or in matrix form as 



^ a I _ J / eit J stiff 
f(e) 
J EAS 



AaW = • (3-78) 



89 



Since the enhancing strain E is chosen to be discontinuous across the element bound- 
aries, it is possible to eliminate the EAS parameter increment Aa (e) at the element level, 
before proceeding to assemble the element matrices into global matrices. Solving for the 
increment Ao w using (3.77) 



Aa (e) - - [*£] ~* (f&s + k%&4#>) , (3.79) 

then substituting (3.79) into (3.76), we obtain the following condensed symmetric element 
stiffness matrix fc$? and the element residual force vector 

= $-{^J T !if' , *& (3-80) 

p (e) _ Ae) _ f (e) , [l.(e)l T ^(e)]- 1 f (e) nRn 

' — J ext J stiff t [« QU J |« QQ J Teas- (3.9 1) 

An assembly of the element matrices fe^ and r (e) leads to the global system 

K T Ad = # , (3.82) 

nel nel 

with -KY = AfeF , # = Ar (e) , (3.83) 

where A denotes the finite-element assembly operator. 

The incremental displacement Ad can be solved by using (3.82), and the displace- 
ment d and d (e) updated. With (3.79), the incremental displacement Ad (e) is used to com- 
pute the increment Aa (e) , which is in turn used to update the EAS parameter a (e) . The 
details of this iterative procedure are provided in the Appendix A.l. 
3.3.4. Material Law in Convected Basis 

For the Saint- Venant-Kirchhoff material, the fourth-order material tensor C is de- 
fined as the second derivative of the stored energy function W s with respect to the Green- 
Lagrange strain tensor E, 

r _ d 2 W s 

L - dEdE > ( 3 - 84 ) 
and the second Piola-Kirchhoff stress tensor S is then expressed as 

dW 

S=-^=C-.E. (3.85) 



90 



The constitutive relation of laminated composites can be described by using an or- 
thotropic material law. For that purpose, we express the components C ljfc ' of tangent elastic 
moduli tensor C relative to the fiber reference axis {ai, a 2 , a 3 } of a lamina, and arrange 



these components in a matrix 



fei 



(see, e.g., Reddy [1997, p.41] and Figure 3.4), using 
the same ordering of the strain components in column matrix form as in (3.56) (see also 
(3.62)). 

b 2 




b 3 — a 3 



Figure 3.4. A fiber-reinforced lamina and fiber reference axes {a^ a 2 , a 3 }. 



(Ollll £<1122 q £1133 

£>1122 £2222 q £2233 

C 1212 

£1133 £2233 q £3333 















£2323 o 

c 1313 



6x6 



(3.86) 



where the components C ljkl take the following expressions 

/Mill £l (1 ~ ^23^32) ^2222 _ E 2 (l- 1/13^31 ) /S3333 ^3 (1 ~ ^12^2l) 

" A ' C - A ' C = A ' 

£1122 #1 (^21 + ^31^23) AU33 _ E 3 Q 13 + U U U 2Z ) - 2233 _ E 2 (>32 + ^12^3l) 

A ".<'-<- A , O - , 



91 



A = 1 - V l2 V 2X - IA>3^32 - ^21^13 T 2^12^32^13 i 

/S1212 _ ^t2323 ^ /M313 /-> 

V — ^12 > ^ — <-*23 ) O — Lri3 , 

= ^ji^i , for (i, j — 1,2,3 , and i ^ j) , 

and Ei,E 2 ,E 3 are the Young's moduli in the principal material directions {ai,a 2 ,a 3 }, 
respectively, and i/y and Gj, the Poisson's ratio and the shear modulus in the {e u e,) plane, 
respectively. Note that, for the special case of isotropy, only two material parameters E 
and v are needed: 

E\ = E 2 = E 3 — E , v\2 — v-n = "is = v , 
Gi 2 = G» = G a = 2(j^TJ) • 

Since matrix C^'J of elastic moduli is associated with the principal material direc- 
tions, we need to transform it from the lamina coordinate axes {ai, a 2 , a 3 } to the global 
Cartesian coordinate axes {bi, b 2 , b 3 }. With 6 being the fiber direction angle relative to the 
global Cartesian system (see Figure 3.4), the relationship between the lamina coordinate 
system and the global Cartesian coordinate system is given by 

a x = cos Obi + sin 6b 2 , a 2 = - sin 9bi + cos 8b 2 , a 3 = b 3 . (3.88) 

Since we are developing the formulation in the convective coordinates associated 
with the basis {G t }, we have to express the tensor C of elastic moduli in the same convec- 
tive coordinates. Thus, 

C = C abcd a a ®a b ®a c ®a d = C m G { ® Oj ®G k ®d, (3.89) 

where the components C abcd are given in (3.86), and the components C ijkl are to be com- 
puted for use with the present solid-shell formulation. 

From the following component forms of the second Piola-Kirchhoff stress tensor S 

S = S ab a a <g> a b * S lj Gi ® G j , (3.90) 



92 

we obtain the relation between the components S ij and S ab as 

S l > = (&-a a )(&-a b )S ab , (3.91) 

where G l 'Gj = $> jt and a* = a\ 

Similarly for the Green-Lagrange strain tensor E, 

E = E cd a c ®a d = EyC? (g) G j , (3.92) 
we obtain the following relation 

E cd = (G k -a c ) (G l -a d )E kl . (3.93) 

Using (3.91) and (3.93) in the following component form of the stress-strain relation (3.85) 
with respect to the basis {a J 

gab m cabcdg^ ■ (3 94) 

we obtain 

5« - (C • a a ) (& ■ m) (<Z* - a c ) (<?' • a d ) C abcd E kl , (3.95) 
which when compared to the component form of (3.85) with respect to the basis {GJ 

& = C^ kl E kl , (3.96) 

leads to 

G«» = (C • a a ) (CP • o 6 ) (G fc • o c ) (G< • a,) &** . (3.97) 

The above relation can also be obtained directly by using (3.89). 

If we expressed (3.97) in matrix form by using the same ordering of strain and stress 
components described in (3.56) and (3.62), which resulted in (3.86), the constitutive matrix 
C ljW J in the convective coordinates associated with the basis {GJ is given by 

[C^] = Tq [&**] T G , (3.98) 



93 



with 



Ta = 



f 2 f 3 



f l f 3 



f 2 f 3 
L 2 L 2 



(3.99) 



(*1) 2 (t\? t\t\ {t\f 

(^2) (^2) ^2^2 (^2) 

2£j^2 ^^1^2 ^1^2 ^1^2 ^^1^2 ^1^2 ^2^1 ^1^2 ^2^1 

(4) (^1) 4^3 (^3) ^3 ^3 

2t\t\ 2t\t\ t\t\ + t\t\ t\t\ + i§^2 ^2^3 + ^3^2 

2t\t\ 2t\t\ t\t\ + t\t\ 2t\t\ t\t\ + t\t\ £^3 + £3^ 

and^ = G j * ai . 

3.3.5. The ANS Method 

The assumed natural strain (ANS) method was originally prepared to relieve the shear 
locking problem that typically arises as the thickness of the shell goes to zero (MacNeal 
[1978] Hughes and Tezduyar [1981], Dvorkin and Bathe [1984]), and was later given a 
mixed variational foundation (Simo and Hughes [1986]). Here we use the ANS method to 
treat shear locking caused by the transverse shear strains and curvature thickness locking 
caused by the transverse normal strain in the present solid-shell element. 
3.3.5. 1. Transverse shear strains 

To avoid shear locking, we adopted the ANS method as applied to the four-noded 
shell element in Dvorkin and Bathe [1984]. Here, a linear interpolation of the compatible 
transverse shear strains E[ z and Efa in (3.12), evaluated at the four midpoints A, B, C, D 
of the element edges, at £ 3 = (see Figure 3.5), is applied 



(3.100) 



where the coordinates of points A, B, C, D are £ A = (0,-1,0), £ B = (1,0,0), £ c = 
(0, 1, 0), £ D = (-1, 0, 0), respectively. 

The above interpolation on the transverse shear strains eliminates the shear-locking 
problem, and allows for pure bending deformation without parasitic transverse shear strains. 



94 



3.3.5.2. Transverse normal strain 

In the case of curved thin shell structures or in the nonlinear analysis, to circum- 
vent the locking effect from parasitic transverse normal strain, we employ an assumed- 
strain approximation for the covariant component E% 3 of the compatible Green-Lagrangian 
strain tensor, as done in Betsch and Stein [1995] for a stress-resultant shell formulation. 
Here, a bilinear interpolation of the transverse normal strains sampled at the four corners 
E, F, G, H of the element midsurface (Figure 3.5) is imposed, that is 

- (3-101) 

2=1 

with Ni = J (1 + $ f 1 ) (1 + £ 2 £ 2 ), and the coordinates of the corner points E, F, G, H 
being C = t E = (-1,-1,0), £ 2 = Z F = (1,-1,0), £ = & = (1,1,0), £ - 
=(-1,1,0). 




Figure 3.5. Eight-node solid shell element in isoparametric coordinates: Sampling points 
for ANS interpolations for transverse shear strains (A, B, C, D) and for transverse normal 
strain (E, F, G, H). 

3.4. Interpolation of the Enhanced Strains 
In this section, we first review the regular enhanced-strain method (Klinkel et al. 
[1999]) and establish the optimal number of internal parameters for the enhancing strains 



95 

in the present solid-shell element to pass the membrane patch test and the out-of-plane 
bending patch test. We then propose a new efficient way to enhance the strains, and prove 
the equivalence of the 2-D plane elasticity elements of Simo and Rifai [1990], Taylor et al. 
[1976] and our new enhancing formulation. 
3.4.1. The Regular Enhanced Strains Treatment 

To include the constant stress in the element (e), the orthogonal condition of EAS 
must hold in (3.20), that is 

J SlEdV = 0. (3.102) 

We define the following component forms of the enhancing strain tensor E as 

E = EnG* (£) ® G j (0 = t kl G k (0) ® G l (0) , (3.103) 

where the enhancing strain components with respect to the covectors G { (£) at any arbitrary 
point £ are denoted by E ijt while those with respect to the covectors G k (0) at the element 
center £ = by E kl . From (3.103), E {j can be expressed in terms of % kl as follows 

E l3 - % [Gi (£) >G k (0)] [G l (0) 'Gj (£)] , (3.104) 
where the covector G k (0) can be computed from the vector G t (0) by 
G k (0) = G*'G,(0), 
with [G kl ] m [Gom]' 1 and G ok i = G k (0) -G t (0) , (3.105) 
The matrix form of (3.104) is 

where the components of the enhancing strain are arranged in the same order as in (3.56), 
and T is the matrix that transforms the strain components relative to the basis {G t (0)} to 
those components relative to the basis {Gi (£)}. 



96 



Using the Column-matrix form {S lj } for the stress components, as in (3.62), and 
using (3.106), we can rewrite (3.102) as 

J {S^} T {E tJ } dV = J {^} r r o {fyW-0. (3.107) 
For constant stresses, we have the following condition on j-Ey j 

J {%j}dv m £ £ £ jdedede = o , (3.108) 



44 



where J is the determinant of element Jacobian matrix of the mapping from the isopara- 
metric space □ to the initial configuration Bq ] of element (e). Let j^yj be defined by 
using the interpolation matrix M and the element parameter as follows 

{£y}.= ±M(0a«. (3.109) 

Substituting (3.109) into (3.106), the enhancing strain {^V,} can be written as 

= with£ = ir M. (3.110) 

Remark 3.3. In Simo and Rifai [1990] and other papers such as Klinkel et al. [1999], 
the matrix Q involves the calculation of the determinant J Q of the Jacobian matrix evaluated 
at eh element center, that is 

g = jT M. (3.111) 

From our numerical experiments, both expressions for Q (without J as in (3.110) and 
with J as in (3.1 1 1)) led to exactly the same results. We therefore use only (3.1 10) for 
computational efficiency. 

I 



If we only enhance the membrane strains [E n , E22, 2E i2 ], and the transverse normal 



97 



strain E 33 , the transformation matrix T in (3.106) should be presented as follows 



Tn = 



■ W) 2 


{a\f a\a\ 




(<4) 2 


{alf a\a\ 


{alf 


2a\a\ 


2a\a\ a\a\ + a\a\ 


2a\a\ 




{a\f a\al 


{alf 



(3.112) 



where the coefficients a{ are evaluated by 

a? = G f (£)•<?>' (0) , i,j = 1,2,3. 



(3.113) 



The interpolation matrix M should be constructed to satisfy (3.108) for arbitrary 
matrix a^. The selection of M is not unique. In the present solid-shell element, the 
matrix M with the minimum internal parameters of five is in the form of 



M = 





o e, 2 

o o e 1 <e 2 o 

£ 3 



(3.114) 



which is the same as that used in Klinkel et al. [1999]. Our numerical experiments showed 
that the selected M as in (3.114) cannot pass the out-of-plane bending patch test, while 
passing the membrane patch test. 

Remark 3.4. The concept of patch test was first introduced by Bazeley, Cheung, 
Irons and Zienkiewicz [1965] and has since demonstrated to give a sufficient condition 
for convergence (e.g., Irons and Loikkanen [1983], Taylor, Simo, Zienkiewicz and Chan 
[1986], Zienkiewicz and Taylor [1997]). A reviewer pointed out that there is no consensus 
about the necessity of passing the out-of-plane bending patch test for convergence, while 
passing the membrane patch test is necessary for convergence. On the other hand, we show 
in Section 5.4 that the solid shell formulation with Five HAS parameters, which does not 
pass the out-of-plane bending patch test, cannot provide accurate results for problems in- 
volving nonlinear material behavior (in addition to large deformation), whereas the present 



98 

formulation with seven EAS parameters, which does pass the out-of-plane bending patch 
test, provides accurate results. I 

To pass the membrane patch test and out-of-plane bending patch test, the bilinear 
polynomials for the transverse normal strain E 33 are necessary (i.e., the minimum number 
of EAS parameters for E 33 should be three, instead of just one as in (3.1 14)). Therefore, 
the optima] number of EAS parameters should be seven, as shown in the matrix M below 



M = 



t 1 

£ 2 

f 1 £ 2 

£ 3 f 1 ^ 3 £ 2 £ 3 



(3.115) 



A computationally more expensive choice for passing both patch tests is to include 
the trilinear polynomials for E 33 and bilinear polynomials for E n (Bischoff and Ramm 
[1997] and Betsch and Stein [1996]). In this case, the number of EAS parameters is nine, 
with the matrix M as shown below 



M = 



e 1 

o e 2 

o o e e ee o o o o 

£ 3 ?f £ 2 £ 3 fl£2|3 



(3.116) 



The results of our numerical experiments showed that there is little advantage in using 
(3.1 16), since improvements compared to the use of (3.1 15) were insignificant. 

If we enhance all the six strain components [E n , 2E 22 , E l2 , E 33 , 2E 23 , 2E 13 ], the 
interpolation matrix M contains complete sets of polynomials up to the trilinear one, and 
thus corresponds to a set of 30 EAS parameters (Andelfinger and Ramm [1993] and Klinkel 
and Wagner [1997]). In this case, the matrix M is as shown below 



m=[m (1 ',m (21) ,m (22, ) m (3) ] , 



(3.117) 



99 



where the submatrices M (1) , M (21) , M (22) , and M (3) are 



M< 21 > 



M (22) = 



r^ 1 o 














" 






£ 2 
























e 


e o 





















cf 3 
























e 


£ 3 









. o 











o e 


e. 






" 

















ee ' 



























ee 


















































ee 


ee 

























ee 


ee 







r ee 

s s 








o 


o 


o 


o 


o 







ee 


























ee 


























ee 


ee 


























ee 


























ee 


























eee 






















eee 
























eee 






















eee 






















eee 



M< 3 > = 



The corresponding transformation matrix T previously discussed in (3.106) is now ex- 
pressed as follows 



T = 



W) 2 


(«f) 2 


a\a\ 


(a?) 2 


a\a\ 


44 


(<4) 2 




a\a\ 






44 


2a\a\ 


2a\a\ 


a\a\ + a\a\ 


2a\a\ 


a\a\ + a\a\ 


44 + a 2 a l 




a\a\ 






44 


2 a 2 a 3 


2a\a\ 


a\a\ + a\al 


2a\a\ 


44 + a 3°2 


44 + a 3 a 2 


2a\a\ 


2a\a\ 


a\a\ + a\a\ 


2a\a\ 


44 + a 3°i 


44 + a 3 a i 



(3.118) 



where the coefficients a{ are the same as (3.1 13). 

Without a combination with the ANS method to remedy the shear-locking problem, 
the above 30-parameter EAS element (Klinkel and Wagner [1997]) cannot pass the out-of- 
plane bending patch test by itself. 



100 



3.4.2. Proposed Efficient Enhancing Strains 

In the traditional EAS method, as presented above, the 3 x 3 matrix [G ij] has to be 
inverted in each element so as to obtain the covector G l (0). For models of composite struc- 
tures that involve a lot of elements in the thickness direction, such inversion clearly adds 
to the computational cost. Here we propose a new method (for calculating the enhancing 
strain) that avoids the inversion of [C?oy], while still passing the required membrane patch 
test and the bending patch test. Moreover, this new method yields the same performance in 
terms of accuracy when compared to the traditional method that needs the covector G i (0) 
in (3.106), (3.112) and (3.113). 

Similar to (3.92), the enhancing strain tensor E (£) can be expressed with respect to 
either the convected basis {GJ or the Cartesian orthonormal basis {e a } as follows 

E = E {j G* (0 ® G j (0 = e« t k <g> e' , (3.1 19) 

where t k = e k , and the components e ki have the same structure as that of Ey as expressed 
in (3.109). 

With the use of the orthonormal basis {e fc }, we thus avoid the computation of the 
covectors G k (0) and the inversion of a 3 x 3 matrix, as mentioned above. 

To compute the tangent stiffness matrix, we need to compute the enhancing strain 
components By (£) at the Gauss points. Since our formulation is based on convective 
coordinates, we need to express the compatible strain tensor E c and the enhancing strain 
tensor E in the same convected basis, so to add these components together to form the total 
strain tensor E as expressed in (3.18). 

Once the components e ki in (3. 1 1 9) are known, the components Eq % can be computed 
from e kl and {Gj (£)} as follows 

%ii (0 = hi [t*Gi (0} [c l 'G 3 (0] , (3.120) 

where, unlike the use of the covectors ^ (0)} in (3.104), we do not need to invert any 
matrices, since the basis {e'} is orthonormal and thus e* == e\ 



101 

The next question is how to select the orthonormal basis |c fc } for use in (3.1 19) and 
for each element. For the case of fiat plates, the convected basis {GJ can be chosen to be 
colinear with the global Cartesian basis {ej. In this case, we simply choose the basis {ej 
to be the same for all elements such that 

ei = Ci, i= 1,2,3. (3.121) 

For the case of curved shell, in each element (e), we select the orthonormal basis {cj such 
that e 3 is colinear with the convected basis G 3 evaluated at the first Gauss point of the 
element (e), that is 

3 - iwm ■ <3 - i22) 

and d and e 2 are obtained by rotating the basis vectors e x and e 2 through same rotation 
operation that rotates e 3 to coincide with e 3 as defined above. The computation of the 
rotation matrix for the above operation is given in Remark 3.5. It is noted that this rotation 
matrix yields directly the components of the basis vectors ei and e 2 . 

Remark 3.5. Given any two vectors e, t with e + t there exists a unique rotation 
tensor A such that 

e= Ae » (3.123) 

where 

A = (e«e) J 2 + e x e + _L_ ( e x e) ® (e x e) , (3.124) 

where the symbol ' over a vector designates the skew symmetric tensor associated with 
the vector (i.e. having the vector as its axial vector, for more details, see e.g., Vu-Quoc, 
Deng and Tan [2000]). 

Let e be the global Cartesian basis vector e 3 , and e be the basis vector c 3 = c 3 e t =: 
fet as defined in (3.122). The rotation tensor A can then be expressed in terms of the 
components f = e 3 as follows 

A = ej (g) e, such that A »ej = cj , (3.125) 



102 



Let a = U for easy recognition of the symbols. We have 



(3.126) 



With (3.126), (3.125) becomes 



A — ti ® ej = t\ei ® ej , 



(3.127) 



with 



A 5 



With the superscript i in AJ designating the row index, and the subscript j designating the 



column index. We have the expression for 
t l z ei = fei as follows 



A! 



in terms of the components of e 3 = e^e* = 



2 \2 



t 3 + 



1 +1 



t 1 t 



1+ t 3 

t 1 t 2 
1 + t 3 

-t l 



1 + t 3 

t 3 + 



1 + t 3 



t 2 t 3 



(3.128) 



Thus the first column in [A}] contains the components t\ of t x = c x = ^e,. The second 



column of 



A', 



contains the components of t 2 = e 2 = t^. 



3.4.3. Equivalence Between EAS Element and Incompatible Mode Element 

In this section, we will show the equivalence of an EAS four-node element in plane 
elasticity as presented in Simo and Rifai [1990] and the incompatible-node four-node ele- 
ment of Taylor et al. [1976] by using the tensor form. In addition, we will also derive a new 
element formulation and prove that it too leads to a condensed stiffness matrix identical 
to that of the incompatible-mode element of Taylor et al. [1976]. The tensor form of the 
enhancing strain does indeed allow one to see the connection between various formulations 
in an elegant and simple manner. This approach has not been exploited in the literature. 
Note that even though Simo and Rifai [1990] stated that their element is in fact identical 



103 



to the incompatible-mode element of Taylor et al. [1976], they provided no proof, which 
is not immediately obvious, even though numerical experiments did confirm that the two 
elements are identical. Moreover, we have not seen any such proof of equivalence in the 
literature, to the best of our knowledge. In this section, we are mainly concerned with the 
small strain case, and thus the small strain notation e is used throughout the section. 
3.4.3.1. Tensor form of enhancing strains 

The second-order enhancing strain tensor e can be expressed as 



where IE is a fourth-order interpolation tensor containing the polynomial basis functions, 
and a a second-order tensor containing the EAS parameters of an element. 

The enhancing strain e can be expressed in either the basis {G a (0)} or the basis 
{e a } as follows 



« (0 = tap (0 G a (0) <g> G" (0) = e ab (£) e a ® e b , a, /?, a, b = 1, 2 , (3. 130) 



where £ a/3 are the components of e with respect to the convected basis {G a (0)} evaluated 
at the element center, and e ab the components of e with respect to the global Cartesian basis 
{e a = e a }. Also, note that indices a and a in (3.130) take values in {1, 2}, since we are 
dealing with 2-D elements here. 

Using (3.130), we obtain the following relationship between components <£ a0 and 
components e a b of e 



(3.129) 



£«0 = £ab [e a 'G Q (0)] e b 'G (0) 



(3.131) 



or in matrix form 




(3.132) 



104 



with the strain matrix {<E Q/3 } = [g n , g^, 2e 12 | T and the matrix F„ defined below 



F T = 



(a\) 2 (a 2 ) 2 



a}a 2 



a 2 a 2 



(« 2 ) 2 (^) 2 

_ 2a\a\ 2a\a\ a\a\ + a\a\ 
where the coefficients a£ are obtained by 

a£ = G Q (0)-e^, a,/9-l,2 



(3.133) 



(3.134) 



Similarly, the EAS parameter tensor a can also be expressed with respect to different 
bases. Here, we choose to express a with respect to the basis G m (0) ® e n and to the basis 



e 9 as follows 



a^a«G^(0)®e" = a<y®e«, 



(3.135) 



where the superscript (s) in the components aj>j, of a with respect to the basis G m (0) ® e" 
represents the EAS parameters in the Simo and Rifai [1990] formulation, and the super- 
script (t) in the components af g of a with respect to the basis ® e» represents the EAS 
parameters in the Taylor et al. [1976] formulation. 



From (3.135), the transformation between the components qW and af n is then 



or in matrix form 



«a=4«( c/,Gr m(0)) . 



(3.136) 



(3.137) 



where the indices (mn) and (fn) are arranged in the order {11, 22, 12, 21}, and the matrix 
M~ 1 take the expression 



M" 1 



2 1 



a{ a 2 

a 2 , al 

a 2 a} 

a\ a 2 



(3.138) 



105 



where the coefficients were given in (3.134). 

For the relation between the enhancing strain components £y and the EAS parame- 
ters qW, it follows from (3.129), (3.130)i and (3.135)! that the fourth-order interpolation 
tensor E can be expressed in component form as follows 



E = E^.-G* (0) <g> G j (0) ® G k (0) g e, , 
leading to the following component matrix equation 



(3.139) 



= #C {4?} 



(3.140) 



Similarly, for the relation between the enhancing strain components e ab and the EAS 



parameters af g , it follows from (3.129), (3. 130) 2 , and (3.135) 2 that the interpolation tensor 
E can be expressed in component form as follows 



E = E {t)C a d b e a <g> e b <g> e c ® e d , 



(3.141) 



leading to the following component matrix equation 



{eat} = {««} . 



(3.142) 

Substituting (3.137) into (3.140), then using the result in the left hand side of (3. 132), and 
next substituting (3.142) into the right hand side of (3.132), we obtain the relation between 



and 



E {t) 



as follows 



E {s] 



ki 



E^ 



M . 



(3.143) 



The interpolation matrix 



E {t) 



is chosen to be the same as in Taylor et al. [1976], 



and the derivatives evaluated at the element center, that is 



-tfy* (o) o ^y,v (o) o 

o £ l x#{0) o -£ 2 x $ i(o) 

e i x 4a (o) -e y , e (o) -ex <e (o) w 



(3.144) 



106 



where the rows correspond to the strain components {e n , e 2 2, ^12}, and the columns cor- 
respond to the EAS parameters a£j arranged following the order erf = {11, 22, 21, 22}. 
By substituting (3.144) into (3.143), and by rearranging the columns in the resulting 



component matrix 



J(W|J 



, we obtain 10 



E {s) 



f 1 


e e 



(3.145) 



where the rows correspond to the strain components j(£ n , <E 2 2, 2<Ei 2 }, and the columns cor- 
respond to the EAS parameters aft arranged following the order kl = {11, 22, 12, 21}. It 
can be seen that (3. 145) is the same strain-enhancing interpolation matrix for 2-D elasticity 
elements as suggested in Simo and Rifai [1990]. 

Using the tensor formalism, we can derive an EAS formulation that is different from, 
but equivalent to, the EAS formulation in Simo and Rifai [1990] and the incompatible 
element of Taylor et al. [1976]. To this end, let's express the interpolation tensor E with 
respect to the basis vectors {d (0)} and {G i (0)} at the element center as follows 

E = (0) <g> G j (0) ® G k (0) ® G, (0) . (3. 146) 

In parallel to the above, let's express the EAS parameter tensor a in the same basis: 

a = aWGi'(0)®G*(0) . (3.147) 

Similar to (3.135), we find the relation between qJJ and in matrix form to be 

{aW}=T- 1 {4?}, (3.148) 

where the indices (pq) and (kl) follow the order {11, 22, 12, 21}, and the matrix T -1 takes 
the form 



r „i 



a[ a{ 

a 2 2 a\ 

a\ a\ 

a? a} 



(3.149) 



Symbolic computation was used to carry out the computation in (3.143) to obtain the result shown in 
(3.145). 



107 



when the coefficient are computed as in (3.134). 

Since the enhancing strain j<£ij} with respect to the convected basis {Gi (0)} can be 
expressed as 



{«»} = K>r;] «>} 



E {s) 



ki 



K?} , 



(3.150) 



it follows from (3. 148) and (3. 150) that the relation between the interpolation matrix 



and the interpolation matrix 



E^ Pq 

13 



is given by 



E {s) 



(3.151) 



E 



kl 



that 



There are thus infinitely many ways to define the interpolation matrix 
are equivalent to each other from the tensor viewpoint. The enhancing-strain component 
matrix {e ab } relative to the global Cartesian basis {e a } can then be expressed in several 
ways 

{s ab } = 0«{a&}, with 0® = {tity = F» T M- 1 , (3.152) 

{s ab } = W {<*«}, witha (s) = F - T {jB^} = gWM, (3.153) 

{ea*} = a w {«a}. witha W = F - r {lE(^} = a (8) T ) (3.154) 

where the matrices £ (t) and (? (s) are the interpolation matrices relating to the EAS parame- 
ters a mn and Q £i- respectively, to the Cartesian strain components e^, and £ w is another 
choice of interpolation matrix. 
3.4.3.2. Equivalence of condensed stiffness matrice s 

The matrix form of the strain tensors is used in finite-element formulation. In a 
typical element (e), the compatible strain matrix {e c ab } and the enhancing strain matrix 
{e ab } are interpolated respectively as follows 



{e c J = Bd< e > , 
{eab} = S« (e) , 



(3.155) 
(3.156) 



108 



where B is the strain-displacement matrix, and Q one of the interpolation matrices given 
in (3.152)— (3. 154), depending on the formulation used. 

The enhanced strain matrix {e a6 } is obtained by adding the enhancing strain {e ab } to 
the compatible strain {el b } 

{*<*} = + = B | J = [B 9] I ^ } . (3.157) 
The element stiffness matrix fc (e \ which is similar to (3.78), can be obtained by 

= J B T cBdv = J | » )c[sg]dv 

B M -(«) ^ ' 

/ B T CBdV j B T cgdv 



,(e) 



/ g r CBdv / g T cgdv 



,(e) 



,(e) 



ju(e) ju(e) 
fc(e) fc (e) 



(3.158) 



where C is matrix of linear elastic moduli. 

The condensed element stiffness matrix has the same form as in the formulation by 
Taylor et al. [1976], Simo and Rifai [1990], and in the formulation presented in (3.146)- 
(3.149) 



fc(e) _ u{e) _ .(e) ^(e)]" 1 jr. 



(e) 
"au 



(3.159) 



For the formulation of Simo and Rifai [1990], the second term in (3.159) leads to 



fcto [l,(e)l"'L 



(<0 
au 



J B T cg {s) dv 



j g^ T cg {s) dv 



1* 

*(■=) 



(s) T 



CBdV 



(3.160) 



Substituting in (3.160) the following relation as given in (3.153) 2 

g(s) (0 = git) (0 M ( 



(3.161) 



109 



where M is the inverse of M 1 in (3. 1 38), and is a constant nonsingular matrix. 
Then substituting (3.161) into (3.160), we obtain 



k, 



» _ 



J B cT cg {t) dv 



ft 



MM 



j Q® T CQ {t) dV 



(e) 



M~ T M T 



j Q [t)T CBdV 



= J B T cg {t) dv 



1 -1 



j g® T cg®dv 



J g {t)T CBdv , 



(3.162) 



which is exactly the same as the second term of the condensed element stiffness matrix fc (e) 
in (3.159) obtained from the formulation of Taylor et al. [1976]. Since the compatible part 
k^l of fc (e) are the same, the condensed element stiffness matrix k (e) in Simo and Rifai 
[1990] is exactly the same as in Taylor et al. [1976]. 

From (3.154), we have the following relation between and 

g^(i) = g^(i)T'\ (3.163) 

where T is the inverse of T _1 in (3. 149), and is a constant non-singular matrix. Following 
the same procedure as in (3.159) and (3.162), one can easily establish that the new enhanc- 
ing strain as given in (3. 146)-(3. 149) yields a condensed stiffness matrix identical to that 
obtained by Taylor et al. [1976] and Simo and Rifai [1990]. 

It should be noted that a formulation similar to (3. 146)-(3. 149) but evaluated at an 
arbitrary point £ inside the biunit cube will also produce an element that passes the patch 
test. In the previous section, we have indeed selected £,=£ x (i.e. the first Gauss point). 

3.5. Numerical Examples 

The finite element formulation of the present low-order solid-shell element for static 
analyses of multilayer composite shell structures, presented in the previous sections, has 
been implemented in both Matlab and the Finite Element Analysis Program (FEAP), devel- 
oped by R.L. Taylor (Zienkiewicz and Taylor [1989]), and run on a Compaq Alpha work- 
station with UNIX OSF1 V5.0 910 operating system. The tangent stiffness matrix and the 



110 



residual force vector are evaluated using full 2 x 2 x 2 Gauss integration in each element. 
A tolerance of 10" 18 on the energy norm is employed in the Newton iteration scheme for 
the convergence. Below we present numerical examples involving geometrically nonlinear 
static analysis, with isotropic and orthotropic elastic materials. 
3.5. 1 . Patch Tests and Optimal Number of Parameters 

The patch tests for the membrane behavior and the transverse out-of-plane bending 
behavior of plate and shell elements were suggested by MacNeal and Harder [1985]. In 
these tests, a patch of five plate/shell elements with four external nodes and four internal 
nodes at the (X,Y) coordinates in Figure 3.6. (see also MacNeal and Harder [1985]): 
Since we are interested in submitting the solid-shell element formulation described in the 
previous sections to the above patch test, the number of nodes is actually doubled, with 
one series of nodes at the top surface of the plate (Z = h/2), and one series at the bottom 
surface (Z = -h/2). The aspect ratio is defined by the triplet A := L : W : h, where 
(L, W, h) designates the length, width, and thickness of the plate, respectively. We will 
subject the following elements to the above mentioned patch tests: The proposed solid- 
shell element with seven EAS parameters, the solid element with thirty EAS parameters in 
Klinkel and Wagner [1997], the solid-shell elements with five EAS parameters in Miehe 
[1998/?] and Klinkel et al. [1999]. It is noted that we will discuss the EAS formulation 
based on the displacement gradient in details in Section 4.3, which is more efficient than 
Miehe [1998fc]. 



Figure 3.6. The five-element patch of a plate: Geometric dimension (L : W : h = 0.24 : 
0.12 : 0.001); E = 10 6 , v = 0.25; 1, 2, 3, 4 are node numbers on top surface. 



O 



z 




Coord. (X,Y): 

1 : (0.04,0.02) 

2 : (0.18,0.03) 

3 : (0.16,0.08) 

4 : (0.08,0.08) 



Ill 

3.5. 1 . 1 . Membrane patch test 

The following displacements u, v , w along the X, Y, Z, axes, respectively, are pre- 
scribed at the top and bottom exterior nodes of the plate: 

u = 1(T 3 (X + Y/2) , v = 10" 3 (Y + X/2) , w = . (3.164) 

The theoretical solution is a constant in-plane membrane stress field in all five ele- 
ments of the patch as shown below 

ox - o Y = 1333 , t xy = 400 . (3.165) 

Our numerical results show that the proposed solid-shell element, together with the 
element formulations by Klinkel and Wagner [1997], Miehe [1998ft], and Klinkel et al. 
[1999] all pass the membrane patch test, that is, the computed displacements at the interior 
nodes agree exactly with (3.164), and the computed stresses in all elements agree exactly 
with (3.165). 

3.5. 1 .2. Out-of-plane bending patch test 

To construct a constant stress state, the displacements ( u , v , w Q ) and rotations 
(Ox , Y )at the midsurface of the plate (i.e., Z = 0) should be 

u = v = , w = 10- 3 [X 2 + XY + Y 2 ) /2 , 
e x = 10" 3 (Y + X/2) , 6 Y = -10- 3 (X + Y/2) . (3.166) 

With the deformation as prescribed by (3.166), the displacements of the external nodes at 
the top surface and at the bottom surface of the plate can then be prescribed by 

[u,v,w] T = [«©, vq, u>o] T ±^[#y,-0x,O] T , (3.167) 

which is the prescribed boundary conditions for the out-of-plane bending patch test. 

The exact nodal displacements at the interior nodes are calculated by using (3.167) 
(see, e.g., Table 3.2 for the finite-element displacement of the interior nodes at the top 



112 



Table 3.2. Displacements at internal nodes in bending patch test. Present element with 7 
EAS parameters, and with ANS. 



node number 


u 


V 


w 


1 


2.500 x 10~ 8 


2.000 x 10~ 8 


1.400 x 10" 6 


2 


9.750 x 10" 8 


6.000 x 10" 8 


1.935 x 10~ 5 


3 


1.000 x 10~ 7 


8.000 x 10~ 8 


2.240 x 10~ 5 


4 


6.000 x 10~ 8 


6.000 x 10- 8 


9.600 x 10~ 6 



surface). The theoretical solutions of the stresses at the top and bottom surfaces of the plate 
are 

a x = a Y = ±0.667 , r XY = ±0.200 . (3.168) 

Table 3.3. Displacements at internal nodes in bending patch test. Solid element with 30 
EAS parameters, and without ANS. 



node number 


u 


V 


w 


1 


1.15827 x 10" 7 


2.34981 x 10" 8 


5.44389 x 10~ 6 


2 


1.40031 x 10" 8 


8.58585 x 10" 8 


2.51957 x lO" 5 


3 


1.81490 x 10~ 8 


6.05335 x 10~ 8 


2.89471 x 10~ 5 


4 


1.37490 x 10- 7 


4.70933 x 10~ 8 


1.65964 x 10" 5 



Without the use of the ANS method on the transverse shear strains, the solid element 
even with the full 30 EAS parameters (Klinkel and Wagner [1997]) cannot pass the out- 
of-plane bending patch test, that is, it cannot obtain the same nodal displacements shown 
in Table 3.2, and the stress state shown in (3.168). The calculated displacements at the 
interior nodes on the top surface for the solid element with 30 EAS parameters are listed in 
Table 3.3. 

Some researchers used the ANS method in combination with the EAS method with 
one parameter for transverse normal strain £33 (e.g., Miehe [19986], Klinkel et al. [1999]). 
Here, we show, however, that this formulation cannot pass the out-of-plane bending patch 
test (Table 3.4), while the computed results are much better than 30-parameter EAS solid 
element (Klinkel and Wagner [1997]). These results indicate that the ANS method plays 
an effective role for remedying shearing locking in thin-shell problem. 



113 



In the present work, we propose an optimal formulation for EAS treatment, that is, 
the minimal number of parameters that is required for the element to pass both the mem- 
brane patch test (easy) and the out-of-plane bending patch test (more difficult), in which 
we use both the ANS method and a seven-parameter EAS method (four parameters for 
the membrane strains (E n , E 22 , 2E 12 ), and three parameters for the for transverse normal 
strain E 33 ). It is noted that the three EAS parameters for the transverse normal strain £33 
correspond to a polynomial with one linear term £ 3 and two bilinear terms and £ 3 <!; 2 . 
Numerical results show that this solid-shell element formulation passes the out-of-plane 
bending patch test, namely, obtains exactly the same displacements at the interior nodes as 
shown in Table 3.2, and the stresses at the top surface and at the bottom surface as shown 
in (3.168). 

Bischoff and Ramm [1997] used a nine-parameter EAS method to treat the incom- 
pressibility problem (five parameters for the membrane strains, and the full four parameters 
for the transverse normal strain). It turns out that this choice also passes the membrane and 
the out-of-plane bending patch tests, but with higher computational effort, compared to the 
seven EAS parameters that we are proposing. In this sense, the combination of the ANS 
method and the proposed seven-parameter EAS method (four for membrane strains, and 
three for transverse normal strain) is computationally optimal. 

Table 3.4. Displacements at internal nodes in bending patch test. Solid-shell element with 
5 EAS parameters, and with ANS. 



node number 


u 


V 


w 


1 


2.49406 x 1(T 8 


2.00207 x 10~ 8 


1.39803 x 10" 6 


2 


9.74938 x 1CT 8 


5.94341 x 10- 8 


1.93334 x 10~ 5 


3 


1.00572 x 10- 7 


8.01161 x 10" 8 


2.23496 x 10~ 5 


4 


5.93407 x 10- 8 


5.97675 x 10- 8 


9.55657 x 10" 6 



3.5.2. Cantilever Plate 

We use a single-layer cantilever plate to establish the correctness of the present for- 
mulation, by comparing the computed results to those published in the literature (e.g., those 



114 



from geometrically-exact shell theory (Vu-Quoc, Deng and Tan [2000])). 
3.5.2. 1. Cantilever beam: in-plane bending 

This problem has been previously investigated by Simo et al. [1990] to show the 
superior performance of their mixed finite element shell formulation for the membrane 
behavior based on the Hellinger-Reissner functional. A cantilever beam subjected to an 
end load is discretized with ten elements. The first mesh contains elements with uniform 
and regular geometry (Figure 3.7), whereas the second contains highly distorted elements 
(Figure 3.8). The beam has length L = 1.0, width W - 0.1, and the thickness h = 0.1. 

The material properties are 

£ = 1.0xl0 7 , ^ = 0.3, (3.169) 

where E and v are the Young's modulus and the Poisson's ratio, respectively. 

The load deflection curves for both meshes are shown in Figure 3.9. The present 
solid-shell element based on the Fraeijs de Veubeke-Hu-Washizu functional shows the 
same accurate results and insensitivity to mesh distortion as for the element reported in 
Simo et al. [1990]. 



A 




Figure 3.7. In-plane bending: Deformed and undeformed regular mesh with 10 solid-shell 
elements. 



115 




Figure 3.8. In-plane bending: Deformed and undeformed distorted mesh with 10 solid- 
shell elements. 




0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 

Displacement 

Figure 3.9. In-plane bending: Load deflection curve for both regular mesh and distorted 
mesh with 10 load steps. 

3.5.2.2. Cantilever plate: out-of-plane bending 

A cantilever plate of length L = 10 and width W = 1 is subjected to the transverse 
shear loading F at the free end. We consider three different values of the plate thickness 



116 



h(l, 0.1, 0.01), which correspond to three different aspect ratios L/h(10, 100, 1000), re- 
spectively. Ten elements are used to model this problem (Figure 3.10 and Figure 3.12). We 
use various different aspect ratios to test the performance of the present solid-shell element. 
To have the same level of deflection magnitude regardless of the thickness h, the applied 
loading F is set to be proportional to the thickness raised to power three (i.e., h 3 ) in the 
numerical examples in this section. 

The material properties are prescribed to be 

£=1.0xl0 7 , i/ = 0.4, (3.170) 
where E and v are the Young's modulus and the Poisson's ratio, respectively. 




Figure 3.10. Out-of-plane bending: Geometry and mesh of cantilever plate. 
First, we point out the importance of using a combination of EAS and ANS methods 
in the present element, by solving the linear problem with the transverse loading F = 
10 4 h 3 at the free end. A comparison of the tip deflection of the plate for different options 
in the present solid-shell element and beam theory is shown in Figure 3.11, where all 
results are normalized to the current solid-shell element. There are differences in the results 
obtained from elements with EAS and from elements without EAS (purely displacement 
formulation): the results show that the EAS method clearly improves the bending behavior, 
particularly for low aspect-ratio structures; its influence diminishes dramatically, however, 
as the aspect ratio increases. The ANS method, on the other hand, remains important 
throughout the large range of aspect ratio, from low to high. For aspect ratio larger than 100, 
the ANS method plays a more important role than the EAS method. But the ANS method 
alone cannot provide accurate results, when compared to the exact solutions for beams, 



117 



0.8 
g 0.6 
0.4 
0.2 



-£.----8 1 fr fr 1 ! 



-O disp 

-*- EASonly 

-+- ANS only 

-9- EAS&ANS 

-0- Timoshenko beam 

— E-B beam 



\ \ 
\\ 
w 

W 





io ! 



10 



L/h 



10' 



10' 



Figure 3.11. Out-of-plane bending: Relative importance of EAS and ANS method in re- 
lieving shear locking. 

as shown in Figure 3.1 1. A combination of both EAS and ANS methods is important to 
achieve accurate results, regardless of the aspect ratio. In Figure 3. 1 1 , the small discrepancy 
between the exact solution for the Timoshenko beam and the present sold-shell formulation 
(with EAS and ANS) is due to the effect of the Poisson's ratio. If the Poisson's ratio is set 
to zero, the present solid-shell formulation produces results that agree with the Timoshenko 
beam theory. 

For flat plates undergoing small deformation, numerical results show that it is suffi- 
cient to consider the ANS treatment for only the transverse shear strain E n and E 23 ; the 
additional ANS treatment for the transverse normal strain £33 does not change the numer- 
ical results. 

For the geometrically nonlinear problems, we applied the tip loading in five load 
steps to reach the total force F = 5 x 10 4 /i 3 . The free-tip transverse displacement along 
the force direction at the corner of the midsurface of the plate agrees well with the tip 
displacement obtained from the geometrically-exact shell element (e.g., Vu-Quoc and Tan 



118 



[2002a]) in which the selectively reduced integration was employed; see Table 3.5. Even in 
the extremely thin plate case (aspect ratio = 6667 or h = 1.5 x 10~ 3 ), the present solid-shell 
element yields excellent results, without any sign of shear locking. 

Table 3.5. Out-of-plane bending: Tip deflection of cantilever plate for wide range of aspect 
ratios. Comparison between proposed solid-shell and geometrically-exact shell. 



Aspect ratio Ljh 


Present element 


Geometrically-exact shell element 


10 


7.5083 


7.4897 


100 


7.4146 


7.4144 


1000 


7.4137 


7.4137 


5000 


7.4140 




6667 


7.4139 





To verify the coarse-mesh accuracy of the proposed solid-shell formulation, we use 
the numerical solution obtained from using 640 geometrically-exact shell elements (80 
elements along the length of the plate, and 8 elements along its width) as the reference 
solution. The results tabulated in Table 3.6 show that even the coarsest mesh of two solid- 
shell elements can capture the geometrically nonlinear response with good accuracy. In 
general, the results obtained from the present solid-shell formulation agree well with those 
from the geometrically-exact shell formulation. 



Table 3.6. Out-of-plane bending: Convergence of computed solution, for h = 0.01. 



Mesh (elem. aspect ratio) 


geometrically-exact shell 


present (relative error (%) ) 


2 x 1(500) 


7.61149 


7.61355 (5.6) 


10 x 1(100) 


7.41366 


7.41366 (2.8) 


20 x 2(50) 


7.15862 


7.16073 (0.7) 


40 x 4(25) 


7.19335 


7.19744 (0.2) 


80 x 8(12.5) 


7.20918 


7.21180 (0.0) 



To check the conditioning of the tangent stiffness matrix of the proposed solid-shell 
element, we present in Table 3.7 the number of Newton iterations for each load step and 
the total number iterations for five load steps for three different shell formulation: The 
present solid-shell formulation, the four-node solid-shell of Bischoff and Ramm [1997] 



119 




F 



Figure 3.12. Out-of-plane bending: Undeformed and deformed mesh with 10 solid-shell 
elements. 

(see its implementation in the Appendix A.l), and the geometrically-exact shell element 
(Vu-Quoc, Deng and Tan [2000]). For thick and moderately thin plates (10 < aspect ratio 
< 200), the three shell element formulations have a similar performance. For thin plates 
(aspect ratio > 500), the geometrically-exact shell element formulation provides the best 
performance. The result indicates that the kinematic description of the geometrically-exact 
formulation, in which the displacement of a reference surface and the finite rotation of a 
transverse fiber are unknown kinematic quantities, leads to a better-conditioned tangent 
stiffness matrix than that of the solid-shell formulation, but with more complex operation 
(Vu-Quoc, Deng and Tan [2000], Vu-Quoc et al. [2001]). 
3.5.3. In-plane Bending Problem with Nearly Incompressibility 

As noted before, the displacement-based formulation exhibits severe locking when 
using the full 3-D constitutive models in the incompressible limit. Here we demonstrate 
that the proposed solid-shell element is able to alleviate the parasitic phenomenon. 

The beam is clamped at one end and subjected to an in-plane bending moment at the 



120 



Table 3.7. Out-of-plane bending: Number of iterations in each load step and total number 
of iterations in five load step. 



aspect ratio 


present solid-shell 


4-node solid-shell 


geometrically-exact shell 


10 


9,10, 8, 7, 7(41) 


9, 10, 9, 8, 7(43) 


8, 9, 8, 8, 7(40) 


100 


12,13,10, 9, 9(53) 


12,13,11,10,10(56) 


11,15,10, 10, 9(55) 


200 


13,13,12, 9, 9(56) 


13,14,13,10,10(60) 


12,14,11,10,11(58) 


500 


20,15,16,10,10(71) 


20, 17, 20, 12,10(79) 


13,14,12,10,10(59) 


1000 


25,16,15,10, 10(76) 


26,21,17, 13,11(88) 


15,16,15,11,10(67) 



other (Figure 3.13). The material properties are 

£ = 4.0xl0 3 , v = 0., 0.25, 0.49999, 



(3.171) 



where E and v are the Young's modulus and the Poisson's ratio respectively. 

The beam is modeled with 10 x 4 x 1 finite element mesh with the plane strain 
constraint (constraining the thickness change). The results of this linear problem are listed 
in Table 3.8. The results from the standard displacement formulation, and the classical 
B-bar element (Nagtegaal, Parks and Rice [1974], Simo and Hughes [1986]), and the exact 
solution from the Euler-Bernoulli beam theory for v — are given for the comparison. 



10 



M = 2.75 



0.25 



Figure 3.13. In-plane bending problem with nearly incompressibility: Geometry and mesh. 

Table 3.8. In-plane bending problem with nearly incompressibility: Deflection at center of 
free end 



V 


Displacement 


B-bar 


Present 


Exact 





0.1834 


0.1868 


0.2063 


0.20625 


0.25 


0.17434 


0.1810 


0.1921 




0.49999 


1.591E-04 


0.1453 


0.1456 





It can be seen that the present solid-shell element performs very well for all cases, 



121 



including the incompressible limit, while the displacement formulation and B-bar element 
lags behind for small values of v. The displacement formulation locks severely for the 
nearly incompressible limit in plain strain case. This indicates that for problems where 
the thickness stretch may be constrained (contact or external surface loading) and the in- 
compressible constraints (e.g., plasticity) be involved, the present element will produce a 
reliable results. 

3.5.4. Snap-through of a Shallow, Cylindrical Roof under a Point Load 

This example illustrates the use of the arc-length method (e.g., Simo, Wriggers, 
Schweizerhof and Taylor [1986] and Schweizerhof and Wriggers [1986] for the implemen- 
tation in FEAP) to obtain the unstable static equilibrium response of an elastic shell struc- 
ture that exhibits snap-through behavior. The shell in this case is a shallow, cylindrical roof, 
pinned along its straight edges and loaded by a point load at its midpoint. The dimensions 
of roof and material properties are shown in Figure 3.14. The roof is assumed to deform 
in a symmetric manner, so that one quadrant is discretized, as shown in Figure 3.14. A 
regular 4x4 in-plane mesh of solid-shell elements are used, and two elements are used in 
the thickness direction since the hinged boundary need to be prescribed at the middle line 
of the straight edge of roof. In the R/h = 200 case, the load-deflection path contains two 




Figure 3.14. Snap-through of a shallow, cylindrical roof under a point load. A 4 x 4 x 2 
mesh is used for one quarter of the panel, with symmetric boundary conditions. Two cases 
are considered by varying the thickness: R/h = 200 and R/h = 400. 

limit points, and the displacement control can be used to solve the equations successfully 
(left of Figure 3.15, with 30 load steps). In the R/h = 400 case, the path, however, "kicks 



122 



in" after the first limit point, the arc-length control has to be used (right of Figure 3.15, with 
36 load steps). The results shown in Figure 3.15 are in close agreement with those reported 
in Rifai [1993, p.245], and Example 4.2.6 of ABAQUS [1995]. 




present at central 

□ snell element 

p-esent a: sag* 

Q shell element 




Displacement (mm) 



Displacement (mm) 



Figure 3.15. Snap-through of a shallow, cylindrical roof under a point load. Load- 
deflection path for the R/h = 200 case, where displacement control is employed (left), 
and load-deflection path for the R/h = 400 case, where arc-length control is employed 
(right), both compared to the geometrically-exact shell element. 

3.5.5. Pinched Hemispherical Shell 

The pinched hemispherical shell can be considered as one of the most severe (and 
meaningful) benchmark problem for nonlinear analysis of shells (Stanley [1985]). The 
undeformed configuration of the hemispherical shell has an 18° hole at the top (North 
Pole), and is subjected to two inward forces at 0° and at 180° longitude on the equator, and 
two outward forces at 90° and 270° longitude on the equator, respectively (see Figure 3. 16). 
The material and geometric properties are 



E = 6.825 x 10 7 , u = 0.3 , 



R=10, h = 0.04 , 



(3.172) 



where E and v are the Young's modulus and the Poisson's ratio, R the radius, and h the 
thickness of the shell, respectively. 

Because of symmetry, only one quadrant of the shell is modeled (Figure 3.16). The 
computed displacements for the small-deformation case along the direction of unit loads 



123 



are listed in Table 3.9. For four different meshes with increasing number of elements, 
the values of displacements are normalized with respect to the converged value of 0.094 
(MacNeal and Harder [1985]). The performance of the present solid-shell element is quite 
remarkable, compared to the geometrically-exact shell element. 

Table 3.9. Pinched hemispherical shell: Normalized displacement for linear small defor- 
mation. 



Node per side 


present element 


geometrically-exact element 


3 


1.083 


0.909 


5 


1.040 


0.993 


9 


1.003 


0.987 


17 


0.995 


0.988 



For the large-deformation case, we choose a smaller thickness of h = 0.01 and the 
radius-over-thickness ratio R/h = 1000. The same problem was considered by Parisch 
[1995] for investigating the behavior of several types of elements in thin-shell applications. 
The mesh is composed of 16 x 16 x 1 solid-shell elements. The total load is applied in 
fifteen equal steps. The final deformed mesh configuration is shown in Figure 3.18 without 
any magnification of the deformation. A plot of the pinching loads versus the deflections 
at the corresponding pinching points is shown in Figure 3.17, by comparing with the 4- 
node degenerated shell element in Parisch [1995]. From Table 3.10, it is observed that 
the present solid-shell element is somewhat better than the four-node degenerated shell 
element in Parisch [1995], when both are compared to the converged results of eight-node 
degenerated shell element in Parisch [1995]. 

Table 3.10. Pinched hemispherical shell: large-deformation displacements due to pinched 
force F = 5. 



Element type 


u at B 


u at C 


4-node shell elem. (Parisch [1995]) 


3.24803 


5.43434 


present solid-shell elem. 


3.26055 


5.48331 


8-node shell elem. (Parisch [1995]) 


3.32798 


5.84238 



124 




Figure 3.16. Pinched hemispherical shell: One quadrant of hemisphere with 18° hole. 



3 









] 1 


r 1 1 




point B (outward) 






— point C (inward) 


o x 




o Parisch [1995] 
x Parisch [1995] 


/ / 
/ / 
/ / 
/ / 






P * 






/ / 
/ / 

1 / 






/ / 
d x 
/ / 
/ / 
/ / 
/ / 
/ /* 






/ ✓ 

s 




or ^x- * 
" i i 


s 

s 

1 1 1 



)L^=^_ 1 1 1 1 1 — 

0.1 0.2 0.3 0.4 0.5 

Normalized displacement, u/R 



Figure 3.17. Pinched hemispherical shell: Load deflection curve of the nonlinear calcula- 
tion. 



125 



C 



B 




Figure 3.18. Pinched hemispherical shell: Deformed hemisphere at F/2 = 2.5, viewing 
through hole. 

3.5.6. Multilayer Composite Plate 

While the results in the previous section are restricted to single-layer shell, we now 
provide numerical examples related to multilayer composite shells in both liner and non- 
linear deformation regimes. 
3.5.6. 1 . Two-layer composite plate: linear solution 

Consider a two-layer laminated plate with angle ply (±9) construction (Figure 3.19, 
0° along axis X), are clamped on all sides, and subjected to an uniformly distributed trans- 
verse downward load on the top surface. The side length of the square plate is a = 20.0, 
the layer thickness y)h = 0.01, and the total thickness h = 0.02. The magnitude of the 
uniformly distributed load is q = 1.0. The layer material properties are 

E n = 40 x 10 6 , E 22 = E 33 = 10 6 , 

fi2 = ^13 = ^23 = 0.25, (3.173) 

G\2 — Cn3 = £?23 = 0.5 x 10 6 , 



126 



Because of fiber-orientation induced stretching/bending coupling, which eliminates the 
symmetry condition found in single-layer homogeneous plates, the entire plate has to be 
modeled. A mesh of 6 x 6 x 2 element, with one element per layer in the thickness direction, 
is used. 

In Table 3.11, the transverse displacement of the plate center is compared to both 
the series solution given by Whitney [1969] and the computational results obtained with 
a high-order hybrid multilayer shell element with the same in-plane mesh in Spilker and 
Jakobs [1986]. The present solutions are more accurate than those in Spilker and Jakobs 
[1986] , when taking the series solution of Whitney [1969] as reference. Both sets of nu- 
merical results (Spilker and Jakobs [1986] and Whitney [1969]) show an increased relative 
error when compared to the series solution as the ply angle 9 decreases, with our solution 
being always closer to the series solution of Whitney [1969]. The magnified deformed 
configuration of the two-layer composite plate is shown in Figure 3.20. 



Table 3.1 1. Two-layer composite plate: linear transverse displacement at plate center. 



Angle ±9 


series solution 


Spilker et al. 


present element 


relative error (%) 


±45 


57.80 


58.58 


58.92 


1.95 


±35 


55.26 


56.88 


56.75 


2.71 


±25 


47.10 


51.44 


50.22 


6.62 


±15 


33.82 


40.18 


38.15 


12.8 



To test the performance of the present solid-shell element for high aspect ratios by 
decreasing the plate thickness, we decrease the magnitude of the loading as the cube of 
the plate thickness, so that the transverse displacement at the plate center remains the same 
in the series solution. The results compiled in Table 3.12 show that the computed linear 
solutions are accurate for a large range of aspect ratios, for the ply angle 9 = ±45°. 
3.5.6.2. Multilayer composite plate with ply drop-offs 

This example demonstrates the applicability of the present solid-shell formulation to 
analyze composite structure with ply drop-offs; an example of such structures would be a 
(composite) plate with piezoelectric patches at the top or bottom surface. 



127 




Figure 3.20. Two-layer composite plate: Deformed shape with solid-shell elements. 

Table 3.12. Two-layer composite plate: Transverse displacement at plate center for large 
plate aspect ratios, with ply angle ±45. 



Layer aspect ratio (a/ (e)h) 


series solution 


present element 


relative error (%) 


1000 


57.80 


58.92 


1.95 


10000 


57.80 


58.91 


1.93 


20000 


57.80 


57.54 


0.44 



In this example, each layer is made of unidirectional fiber-reinforced material, with 
the fiber directions aligned at 45/-45/45/-45/45/-45 degrees with respect to the length di- 
rection (Figure 3.21, 0° along axis X). The plate, with length L = 12 and width W = 6, 
has a total of six layers at the thick end, which is clamped; the free thinner end is sub- 
jected to a transverse normal load distribution uniformly along the free edge. The location 
of the ply drop-offs are at X = 4 and X = 8 with the top two layer removed after each 
drop-off. The layer material properties are E n = 25 x 10 9 , E 22 = E 33 = 10 9 , u n - 
fia = ^23 = 0.2 , G n = Giz = G 23 = 0.5 x 10 9 . Three different values of the 



128 

thickness of any given layer (all six layers have the same thickness) are considered, that is, 
(i)h = 0.1, 0.01, and 0.004, for I = 1, 6, where (£) represents the layer number. 

The FE mesh is composed of 288 elements with 12 elements along the length, six 
elements along the width, and one element for each layer through the thickness direction. 
The applied load on the free tip is increased in five load steps up to the total force F = 
6 x 10 9 (t)h 3 , which is proportional to the cube of the layer thickness. The computed free- 
tip transverse displacement along the force direction at the corner of the bottom surface 
of the plate are presented in Table 3.13. It is observed that unlike the isotropic plate in 
Subsection 3.5.2.2, the level of deflection magnitude is not proportional to the cubic of 
the thickness in this nonlinear composite plate problem. The deformed plate is shown in 
Figure 3.22. 




Figure 3.21. Multilayer composite cantilever plate with ply drop-offs: Undeformed mesh 
with = 0.1. 

Table 3.13. Multilayer composite cantilever plate with ply drop-offs: Nonlinear transverse 
displacement. 



Layer aspect ratio Lj ((e)h) 


Transverse disp. 


120(0.1) 


6.72325 


1200(0.01) 


6.11453 


3000(0.004) 


6.01374 



To test coarse-mesh accuracy, we consider the plate with layer thickness mh = 0.01, 
and use the computed solution obtained from the FE mesh with 24 elements along the 
length and 12 elements along the width as the reference solution. From Table 3.14, it can 
be seen that the coarse mesh with six elements along the length and three elements along 



129 



the width already captures the geometrically nonlinear response with a great degree of 
accuracy. 

Table 3.14. Multilayer composite cantilever plate with ply drop-offs: Performance at coarse 
mesh, (t)h = 0.01. 



Mesh (elem. aspect ratio) 


present element 


relative error (%) 


6 x 3(200) 


6.03514 


0.02 


12 x 6(100) 


6.11453 


1.50 


24 x 12(50) 


6.02229 


0.00 




Figure 3.22. Multilayer composite cantilever plate with ply drop-offs: Deformed mesh with 
w h = 0.1. 

3.5.7. Multilayer Composite Hyperbolical Shell with Ply Drop-offs 

This examples was considered to test the current solid shell element formulation in 
shell structures having discontinuous geometry and strong geometric nonlinearity. The 
shell structure consists of three layers with the same thickness — h/3 placed sym- 
metrically with respect to the middle surface and two ply drop-offs at Z = 9 and Z = 15, 



130 



respectively (Figure 3.23). For the shell without ply drop-offs, we have compared with 
Basar, Ding and Schultz [1993] and the results agree each other, while a layerwise shell el- 
ement with complex rotation update was employed in Basar et al. [1993]. Only one eighth 
of the shell structure is modeled with a mesh of 14 x 14 solid-shell elements for inner layer, 
14 x 10 middle layer, 14 x 6 outer layer by assuming the symmetry (Figure 3.23, 0° along 
circumferential direction). The layer material properties are E n — 40 x 10 9 , E 22 = 
E 33 = 10 9 , i/ 13 = u u = i/ 23 = 0.25 , G i2 = G13 = G 23 = 0.6 x 10 9 . The anal- 
ysis was carried out for three different stacking sequences: [0°/90°/0°], [90°/0°/90°] and 
[— 45°/0°/45°]. The deformed shapes shown in Figure 3.24 exhibit a considerable influence 
from the stacking sequence. The shell with ply drop-offs with the [— 45°/0°/45°] stacking 
sequence has larger deformation, and is less resistant to the loading than the shell with ply 
drop-offs with the [0°/90°/0°] and [90°/0°/90°] stacking sequences. The deformed shapes 
in Figure 3.25 for the final load P = 140KN demonstrate clearly that large rotations and 
displacements are involved in this example. 




Figure 3.23. Pinched multilayer composite hyperbolical shell with ply drop-offs: Unde- 
formed mesh. 



131 



so 



120 



100 



40 



60 










Displacement 



-6 



-4 



-2 



2 



4 



Figure 3.24. Pinched multilayer composite hyperbolical shell with ply drop-offs: Load- 
displacement diagrams, v(B) is the displacement along axis Y at point B, u(A) the dis- 
placement along axis X at point A. 




Figure 3.25. Pinched multilayer composite hyperbolical shell with ply drop-offs: Deformed 
shape with stacking sequence [0°/90°/0°] (left) and [-45°/0°/45°] (right). " 




CHAPTER 4 

OPTIMAL SOLID SHELLS FOR NONLINEAR ANALYSES OF 
MULTILAYER COMPOSITES : DYNAMICS 

4.1. Introduction 

The formulation and implementation of the solid shell element for the dynamic anal- 
ysis of flexible multilayer shell structures undergoing large deformation and large overall 
motion is addressed here. In the present formulation, the dynamics of the motion of multi- 
layer shells is referred directly to an inertial frame, thus simplifies considerably the inertia 
operator. The starting point is the nonlinear dynamic weak form based on the internal en- 
ergy and kinetic energy of shells. A linearization of this nonlinear dynamic weak form is 
performed for use in the solution for the kinematic quantities via Newton's method. Time 
discretization is introduced to obtain the time-discrete weak form. A time-stepping al- 
gorithm based on the energy-momentum (EM) conserving algorithm (Simo, Tarnow and 
Wong [1992], Kuhl and Crisfield [1999]) is employed in the time-discrete weak form. This 
algorithm preserves the total momentum and the total energy exactly. Compared to the 
classical Newmark algorithm (Newmark [1959]), the energy-momentum conserving algo- 
rithm gains a more robust stability behavior. On the other hand, a high-frequency algo- 
rithmic dissipation is still desirable to incorporate in the EM algorithm for the stability. A 
number of the modifications of the Newmark family of algorithm have been proposed, typ- 
ically in the form of linear multi-step methods, which introduce high-frequency dissipation 
and preserve second order accuracy (see e.g., Hilber, Hughes and Taylor [1977], Wood, 
Bossak and Zienkiewicz [1981], Bazzi and Anderheggen [1982]). A small modification 
to the present energy-momentum conserving algorithm, in a form similar to generalized- 
a method (Chung and Hulbert [1993]), results in high-frequency dissipation. Numerical 
simulations show that the effect on the transient response is minor for small amount of 

132 



133 



numerical dissipation. 

Since the kinematic description is the displacement of the top and bottom surface 
of the shell in the present formulation, the special treatment on the rotational degrees of 
freedom (Simo and Vu-Quoc [1988], Vu-Quoc et al. [2001]) is not needed. Unlike the 
stress-resultant shell formulation, the resulting consistent mass matrix of the present ele- 
ment is symmetric and independent from the configuration. 

The present eight-node solid-shell element relies on a new optimal (minimum) seven- 
parameter EAS-expansion together with the Assumed Natural Strain (ANS) method that 
passes both the membrane patch test and the out-of-plane bending patch test. In the for- 
mulation, the transverse normal strain and the membrane strains are enhanced by the EAS 
method; and ANS modifications on both the compatible transverse shear strains and the 
compatible transverse normal strain are employed to eliminate the locking effects from the 
compatible low-order interpolations. 

For the EAS approach using enhancing deformation gradient, we develop an EAS 
expansion by superposing the enhancing converted basis to the compatible converted basis, 
and then present a formulation that is much simpler than that employed in Miehe [1998^]. 

The presentation of the present chapter is as follows: we devoted Section 4.2 to 
the dynamic aspect and the use of the energy-momentum algorithm for elastic materials. 
A variant of the EAS formulation based on the deformation gradient (instead of Green- 
Lagrange strains) for solid shells is the focus of Section 4.3. In Section 4.4, several implicit 
direct integration methods with/without numerical dissipation are then used and compared 
in terms of the accuracy, stability and cost in analyses of multilayer shell structures. 
4.2. Dynamics of Solid Shells by an EM Conserving Algorithm 

To obtain a fully discrete formulation, the temporal and spatial discretization of the 
weak form need to be applied subsequently. The readers are referred to Section 3.3 for 
the finite-element discretization of the static weak form. The classical Newmark family of 
implicit integration scheme (Bathe [1996]) is widely used for the temporal discretization. 
The Newmark algorithm is unconditionally stable for linear problems, but only condition- 



134 



ally stable for nonlinear problems (e.g., Simo, Tarnow and Wong [1992]). Since the stabil- 
ity of the time integration scheme in nonlinear problems is related to the conservation of 
energy, Simo, Tarnow and Wong [ 1 992] proposed the energy-momentum conserving algo- 
rithm in an attempt to obtain numerically stable and accurate long-term response. On the 
other hand, the Newmark family are symplectic and momentum preserving, and often have 
excellent global energy behavior with smaller time step (Kane, Marsden, Ortiz and West 
[2000]). In this section, we will present some results of dynamic analyses of solid shells 
using both energy-momentum conserving algorithm (Simo, Tarnow and Wong [1992] and 
Kuhl and Ramm [1999]) and classical Newmark method. 
4.2.1. Time Discretization on Dynamic Weak Form 

N 

The time interval of interest [0, T] = (J [t n , i n+1 ] is subdivided in N time steps. The 

n=l 

state variables such as displacement u n , velocity u n , and acceleration u n at the beginning 
of the time interval [t n , t n+i ] are assumed to be known. The state variables u n+1 , u n+1 , u„ +1 
at the end of the time step [t n , t n+ i] are calculated by a time-discretization algorithm. 

A class of time-discretization algorithms can be presented altogether as a general- 
ization of the Newmark method as follows. Let the inertial (mass) term be evaluated, not 
at the time station t n+l , but at the time t n+CXin = a m t n+ i + (1 - a m )t n , which is a point 
inside the interval [t n , £„+i], with the internal force and external force evaluated at another 
time point t n+a/ ~ a f t n+l + (1 - a f )t n inside the interval [t n , t n +i]- The time-integration 
parameters t n+Ctrn and t n+af have their subscript "m" and "f" being mnemonic for "mass" 
and "force," and are positive numbers between zero and one. The classic Newmark algo- 
rithm corresponds to a f = a m = 1 (Hughes [1987]); the Bossak-a algorithm corresponds 
to Of — 1 (Wood et al. [1981]); the energy-momentum conserving algorithm corresponds 
to a; = a m = i (Kuhl and Ramm [1999]). 

The dynamic weak form at time t n+ct; is 

SU (u n+af ,E n+af ) = j (SE c n+a/ : S n+a/ + 6E n+a/ : S n+a/ ) dV 

Bo 



135 



- / 6u n+aj - (b* n+af -u n+am ) pdV - J 5u n+af -t* n+Qf dS = . (4.1) 

Bo S„ 

The displacement u n+a/ at the time point t n+0lJ and the acceleration u n+otm at time 
point t n+am are generated by a convex combination of the corresponding quantities at time 
t n and time t n+ i as follows 



u n+af = a/u n+ i + (1 - cx f )u n , (4.2) 



u 



n + Qm 



= a m u n+1 + (1 - a m )5I n . (4.3) 



The variation 5u n+a from (4.2) 

<5tt„ +Q/ = ctfSu . (4.4) 

The velocity w n+1 and the acceleration il n+1 at time t n +\ can be expressed as a function 
of the displacement u n+l at £ n+1 and the computed state variables at time t n by using the 
Newmark method, that is 



l = ^2 ( U «+l - «») - - \Jp ~ 1 ] ^ n , (4.6) 



where and 7 are the parameters in the Newmark method, and At = t n+1 — t n the time 
step size. The compatible-strain tensor E° and the enhancing-strain tensor E at time t n+0tf 
are 

K+a, - & («»+«/) . (4-7) 
£7 B+a/ = £?(a n+a/ ) , (4.8) 

where the EAS parameter oc n+a/ is a convex combination of a n and a„ +1 in the same 
manner as that for the displacement u n+0tf as given in (4.2). 

For linear elastic materials, the above convex combination for the compatible strain 
En+a, an d the enhancing strain E n+a/ naturally leads to the identical convex combination 



136 



in the algorithmic stress S n+Qf as follows 

S n +a f = tt/Sn+1 + (1 ~ <Xf)S n 

= CI [a J {E c (u n+1 ) + E{a n+1 )} + (l-a f ){E c (u n ) + E(a n )]\ , 

(4.9) 

which is an essential point in energy-momentum conservation algorithm, see Remark 4.2 
on the conservation of energy and momentum where the need for the definition of S n + a j 
becomes clearer. 

Substituting (4.3) into (4.1), it becomes 

/. T \ 

<^n (u n+Ctf , E n+Qf J = / (5E„ +af I S n+a/ +SE n+af : S n+a/ ^ dV 



+ 



Oir 



pAt 2 



(tin+i - u n ) 



+ I- J 5u n+af -b* n+a/ pdV - J 5u n+af -t* n+af dS 

\ Bo S„ 

= 5U sti ff + 5U mass + SU ext = , 



) 



pdV 



(4.10) 



where 5Tl sti ff is the virtual work generated by the internal forces, SU mass the virtual work 
by the inertia forces, and 6U ext the virtual work by the external forces. 
4.2.2. Linearization of Dynamic Weak Form 

Within a typical element (e), the variation of the internal energy U stl ff in (4.10) at 
time t n+af is 



011 stiff 



BP 



(4.11) 



where the variation of the strain components £?y in matrix form is given by 



(4.12) 



137 



where the strain-displacement matrix B n+0l} is a function of the discrete displacement 
dnla r whereas the interpolation matrix Q given in Section 3.4 is time independent. Note 
that the variations 

<fa£+ a/ = a f Sa {e) . (4.13) 

With (4.12), the variation of the internal energy II ^ in (4.1 1) in matrix form be- 
comes 



511 



- tdL%f% + 6* { :Z f f$ AS , (4.14) 

where f% ff and f$ AS are respectively the internal force related to the displacement S e) 
and the internal force related to the EAS parameter a^. 

The linearization of (4.14) with respect to the nodal displacement d^+i and internal 
parameter aj^ at time t n+1 , with increment Ad {e) = d^ +1 - d { *\ and Aa (e) = a (e) 



n+1 



respectively, is then 

+ (*2Ad« + fcW Aa W) , ( 4.15) 

where the stiffness submatrices fcJ^J, fcj^j, and fc£> are derived from internal forces 

ftug an d , whose expressions were given in (4.14). 

Using (4.2) and differentiate B n+aj with respect to d^ } +l , we obtain 

9B n+a dB n+a dd^at 

= dd (e > dd^ Gaf - (4J6) 

Also, by using the definition of S n+af as stated in (4.9) for elastic material, it follows that 

HS ij } n+a } d{S*} n+1 

flwW = af a» = a f CB n+i ■ and (4.17) 



138 



HS^} n+Qf d{SV} n+1 _ 
a («) af a (e) 



Remark 4. 1 . It is noted that in (4.18), the stress {S lj } n+af should not be computed 
from {Eij} n+af by using the constitutive relation S = C I E, that is 

{^L^^w < 4 - i9 > 

instead, {S tj } n+ should be computed from (4.9). I 
By using (4.16) and (4.17) , the tangent stiffness submatrix fe|*| in (4.15) at time t n+a/ is 

*S = jjgf = <*/ / (G T S„ +Q/ + B T n+af CB n+1 ) dV , (4.20) 



■(«) 




where S>„ +Q/ is the same stress matrix as explained in Section 3.3, but evaluated at time 



Next, by using (4.18), the tangent stiffness submatrix fcjfj in (4.15) at time t n+af takes the 
form 

fcS = ~^ = «/ / Bl +aj CgdV. (4.21) 

Similarly, using (4.17), we obtain the following expression for the submatrix fc£2 in (4.15) 
as 

*&! = = a / / ifCBnHdv , (4.22) 

^"+1 5 /e) 

and using (4.18), the submatrix k£l in (4.15) at time t n+a} as 



df 



M 
EAS 



da {e) 



= a f J g T CQdV . (4.23) 



"+ 1 B (e) 



It can be observed in (4.20) that the two matrices B n+Cc/ and B n+l are evaluated 
at two different configurations c£ e ] a/ and dj+i, respectively, it follows that the submatrix 



139 



fc<2> is non-symmetric. Similarly, we have feg ^ (fc£) T . The consequence is that the 
condensed tangent stiffness matrix kf (see (4.31)) will be also non-symmetric. 
From (4.10), the discrete weak form of the inertia force is 

Ohn ( Je) ,(e)\ _ ^L' d (e) . a " " 2 % e 



- A^ e)r f( fi ) ( 4 - 24 ) 

— UUj n+afJ mass > 

where the element mass matrix is obtained by using the same spatial discretization as 
in Section 3.3 for 6u n+af , u, u, and u: 

m W = J N T NpdV . (4-25) 

The linearization on the weak form of inertia force (4.24) is 

ML-*** = fe™ (e) ) Ad(e) • (426) 

Likewise, the contribution to the weak form (4.10) from the external forces takes the form 

where the element external force f^ t at time t n+aj is 

f% = J ^ + / • (4-28) 

It follows from the above expressions that the linearization of the dynamic weak form 
(4.10) can be written as 

vm%- (Ad"UaW) +D*n£L.-AdW = ^^-OTSL:-«nSS • (4-29) 

Substituting (4.14), (4.15), (4.24), (4.26) and (4.27) into (4.29), for any admissible 5d£ a/ 
and fa$a , we obtain the algorithmic tangent stiffness matrix and the force vector as 
follows 

f A^(e) 1 f -f (e) _ f (e) - f (e) 
iiO \ J ext J stiff J ma 



fc (e) ju(e) 



ext J stiff J mass I (A an\ 



140 



The condensation of the incremental EAS parameters Aa (e) in (4.30) is similar to the static 
case, as presented in Section 3.3. On the other hand, the non-symmetry of the tangential 
stiffness operator as explained above in (4.20)-(4.22), together with the addition of the 
inertia force f^ ass given in (4.24), have to be taken into account. Consequently, the effec- 
tive element dynamic tangent stiffness kft and the effective element residual force are 
computed from (4.30) as follows: 

kV = fcjfj + ,o m ^ — 4a ^aa ^au > (4-31) 



,(e) f ( e ) _ f ( e ) _ f (e) , ju(e) 



-1 



(e) fV) _ f\e) _ f{e) , ju(,e; .(e)] 1 Ae) 

r — Jext T stiff J mass ^ K ua [ K aa\ J EAS ' V*-J*) 

After an assembly process as in Section 3.3, a global matrix equation for the incremental 
displacement Ad is obtained. Once solved, the incremental displacement Ad is used to 
compute the incremental EAS parameter Aa' e ' at the element level , used for the update 
of the EAS parameter ot^ e \ 

With the converged solution d^ +l and ot£\\ obtained at time i n+1 , the acceleration 

and velocity d^]_ x are updated in the classical manner as shown in (4.6). 

Recall that the above general form of time stepping algorithm encompasses the clas- 
sical Newmark algorithm («/ = a m = 1), the Bossak-a algorithm (a/ = 1), and the 
energy-momentum conserving algorithm (a/ = a m = |). 

To check the conservation property of the above algorithms, the total energy and 
the linear and angular momenta have to be calculated. The total energy is the sum of the 
internal energy 8 int and the kinetic energy fC 

£ tot = jC + £ mt , (4.33) 
K = -J pu-udV , (4.34) 

Bo 

g int = -Je: SdV . (4.35) 

Bo 

where the velocity it, the strain E, and the stress S are calculated at each time step t n , and 



141 

so are the linear momentum C and angular momentum J 

C = I pudV , (4.36) 

Bo 

J = J pxxxdV , (4.37) 

So 

where x is the position vector in the current configuration, and is related to the position 
vector X in the initial configuration and the displacement vector u, as given by 

x(&t) = X(£) + u(t,t) , (4.38) 

similar to (3.4). 

Remark 4.2. In the case where otf = a m = | and b* = t* = (no external 
loading), energy and momentum in the system is conserved. The total energy S tot , defined 
in (4.33), varies within the time interval [t n , t n+ i] according to 

Ci - C = Kn + i -Kn + C-'i - SP ■ (4-39) 
From (4.2)-(4.3), it follows that 

U n +\ = \ («n+l + «n) , (4.40) 



For elastic material, that is 



'E c (u) + E (a)] , (4.41) 



s = c : e = c : 

the internal energy S tnt can be rewritten as follows 

g int = J E , SdV = \J E ; C : EdV , (4.42) 

leading to the following expression for the increment of the internal energy within the time 
interval [t n , t n+1 ]: 

£n+i _ z™ 1 = 2 / (E n +i ' c I E n +i — E n : C '. E n ) dV 

Bo 

= -J (E n+1 - E n ) : C : (E n+l + E n ) dV . (4.43) 

Bo 



142 

Let the algorithmic stress S n+ i be defined as follows: 

S n+ i := \ (S n+1 + S n ) = \C I (E n+1 + E n ) , (4.44) 
From Taylor expansion, we expand the strains E n+ i and E n at time t n+ ± to get 

E n+1 -E n = AtE n+ x + O (At 3 ) , (4.45) 
Using (4.44) and (4.45), (4.43) becomes 

Ci - £ n* = ^ / E n+ , : S nH dV + O (At 3 ) , (4.46) 

Bo 

Likewise, it follows that 

u n+1 -u n = Atu n+ x + O (At 3 ) , (4.47) 
and hence the change of the kinetic energy K in the interval \ t n , t n +i ] is 

Kn+i-Kn = - Jp (u n+ i«t£ n+ i -Un* Un) dV 

Bo 

If. 

= -Jp (u n +i + u n ) • (tt n+1 - u n ) dV 

Bo 

= At I pu nH -u n+ i_dV + O (At 3 ) , (4.48) 

So 

Substituting (4.48) and (4.46) into (4.39), it shows 



ctot c tot 

c n+l °n 



J K + i + E n+ i : S n+ i] AtdV + O (At 3 ) 

Bo 

= n n+ iAr. + O (At 3 ) , (4.49) 
If il n+ 1 = 0, it means that the stationary condition of functional IT at time t n+ i, namely 

SU(u nH ,E nH )=0, (4.50) 
therefore the total energy is conserved in the time interval [t n , £ n +i] 

O-C = 0(Ai 3 ) • (4-51) 

I 



143 



4.3. Enhanced-Assumed-Strain Method Based on Deformation Gradient 
In this section, we will present the weak form and the finite -element discretiza- 
tion for the proposed solid-shell element by using the enhanced-assumed-strain (EAS) 
method based on the deformation gradient, instead of the reparametrization of strains in 
Section 3.2." 
4.3.1. Weak Form 

The non-linear version of EAS method by Simo and Armero [1992] is based on the 
decomposition of the deformation gradient F into the compatible part F c and the enhanc- 
ing part F as follows 



F = F c (u) + F . 



(4.52) 



In this case, the three-field Fraeijs de Veubeke-Hu-Washizu functional, depending on the 
displacement field u, the enhancing deformation gradient F, and the nominal stress tensor 
P = FS is given by 

/ \ 



n(u,F,p) = J w s (e (f c ( u ) + f)) dv - J p : Fdv 



So 



I 



+ ijwupdV + f- J u-b*pdV - J u-t' 

\Bo / \ Bo S„ 



dS 



(4.53) 



where U sU ff corresponds to the internal energy, II mass the work by the inertia force, and 
U ext the work by the external forces. 

Remark 4.3. The naming of the stress P in (4.53) is not unique, and can be confus- 
ing. In component form, P is written as 



p = P aA e a ® Ea , such that t = P- N , 



(4.54) 



1 1 The resulting element cannot pass the out-of-plane bending patch test while pass the membrane patch 
test, see the numerical examples of Section 3.5 for details. 



144 



where e a is the basis tangent vector in the current configuration, E A the basis tangent vector 
in the initial configuration, t = t a e a the traction force acting on a facet in the current 
configuration, and N = N A E A the normal of the same facet in the initial configuration. 
Truesdell and Noll [1992, p. 100], Chadwick [1976, p. 124] and Marsden and Hughes [1983, 
p. 135] refer to P as the first Piola-Kirchhoff stress tensor, whereas Malvern [1969, p.222] 
refers to T = P T , namely, the transpose of P, as the first Piola-Kirchhoff stress: 

T = T$ a E A ®e a , such that t = N • T . (4.55) 

Chadwick [1976, p.99], on the other hand, refers to T = P T as the nominal stress. The 
reason for the difference in the naming of this stress tensor lies in the convention adopted 
for the indices in the component of the (Cauchy) stress tensors (Malvern [1969, p.224]), and 
thus the way these tensors operate on the normal to yield the traction vector. Even though 
Chadwick [1976] adopted the same convention for the indices as in Malvern [1969]), he 
defined the traction vector as: 

t = P- JV = Tl • N = N -T , (4.56) 

and called T the nominal stress, and P = T% the first Piola-Kirchhoff stress. Here we 
follow the convention and naming as in Malvern [1969], that is, T = P T being called 
the first Piola-Kirchhoff stress, and thus call P the nominal stress. 

The stress power, which is related to the term / P FdV in (4.53) is written as 

Bo 

follows (Malvern [1969, p.224]) 

J P : FdV = / To • 'FdV , (4.57) 

B Bo 

since 

P : 'f = P aA (F) a A = T Aa (F) a A = T "'F . (4.58) 

The nominal stress P and the first Piola-Kirchhoff stress T = P T is related to the second 
Piola-Kirchhoff stress as follows 

P = Tq = FS . (4.59) 



145 



Remark 4.4. Regarding the functional U mass related to the inertia force, we recall 
that the variation of the displacement it is taken to be independent of time (see (3.21)), and 
thus u s = u, since Su = 0. I 

Using (3.12) and (4.52), the variation of the strain E is given by 

SE = F T SF = F T (<5F C + 5F) , (4.60) 

we have 

dW. 

where — — is a symmetric second order tensor (see Section 3.2). 
oE 

The weak form of (4.53) is expressed as 

SU (u, F, P) = SU stlff + 5n mass + SU exi , (4.62) 

where the weak form 6U sti ff of the internal energy II S ^ is obtained by using (4.52) and 
(4.61) 

r dW r / ~\ r dW 

SU stiff = J-^\ 5EdV - J S(PI F)dV = J F-^ SF c dV 

Bo Bo B 

+ J F^- *. SFdV - J SP : FdV - J PI 5FdV , (4.63) 

Bo Bo Bo 

whereas the weak form 8U mass for the inertia force is simply 



SU mass = JSwupdV, (4.64) 

So 

and the weak form <511 &rt for the externally applied forces 

SRext = ~ J 5wb*pdV - J Swt*dS . (4.65) 

Bo S a 



146 



Again the nominal stress P can be eliminated from the above weak form (4.63) by en- 
forcing the orthogonality condition between P and the enhancing deformation gradient F, 
namely 

J 6P I FdV + JP'» SFdV = . (4.66) 

Bo Bo 

Using the definition of the notation S as in (3.28), the weak form 8U stl ff becomes 

6U stlff = J SI 5EdV = I [FS : 6F C + FS : SF) dV . (4.67) 

Bo Bo 

If we develop the formulation in the convected basis, the deformation gradient F is ex- 
pressed by 

F = F c + F = g i ®G i , (4.68) 

where the additive decomposition of the total deformation gradient F directly leads to an 
additive decomposition of the convected base vectors g { in the current configuration into 
the compatible part g\ and the enhancing part g { , namely 

9l = 9! + 9i, (4-69) 

F c = g1®G\ F = ^®G l . (4.70) 
It follows that the Green-Lagrange strain tensor E, which takes the form 
E = \(F T F-E 2 ) = ±(g i -9 j -G i -G j )G i ®& 

= \ (g\'9) + 9\% + 9i'9 C j + 9i'9i ~ GrGj) G* ® G j 

= E c + E. (4.71) 



where 



E 



= \(9t'9 c j -G i -G j )G i ®& , (4.72) 
E = \(9\%+g i '9) + 9 i %)G i ®G i = \9; j G i ®GP . (4.73) 



147 



The variation of components of the Green-Lagrange strain E is given by 

SEij = i (Sgi'Qj + 9 l 'Sg j ) = sym (Sg^gj) . (4.74) 

Substituting (4.69) and (4.74) into (4.67), the weak form 5U sti ff now reads as 
m stlB = J S^SE l3 dV = J S lj sym (S gi '9j) dV 

= J SPsym (Sg c r9j + Sg^) dV . (4.75) 

Bo 

4.3.2. Finite Element Discretization and Linearization 

Following the same spatial discretization procedure as in Section 3.3, the compatible 
convected basis vectors g c at a point inside an element (e) in the current configuration are 
functions of the displacement d (e) of element (e). 

To pass the patch test, Simo and Armero [1992] suggested the use of the enhancing 
deformation gradient F tensor in the following form 

F = J -jF Q J TJ- Q l , (4.76) 

where J is the Jacobian determinant at a point £ inside an element, and J the Jacobian 
determinant evaluated at the element center £ = . In convected coordinates, the compat- 
ible deformation gradient F Q , and the element Jacobian J , both evaluated at the element 
center, are expressed as 

F Q = g c Ql ®G i , J Q = G 0l ®e\ Jo 1 = e 4 <8> , (4.77) 
where g oi and G 0i are given by 

dx(0) _ _ dX (0) 

9oi ■= — . G oi •= d £ . ( 4 - 7 *) 

and Gq are obtained from G 0i , the vectors e { = e' form the global orthonormal basis. The 
tensor T in (4.76) is the same as in Simo and Armero [1992], namely 

3 

T = £ T; ® GRAD^/V' = T)e x ® e? 7=1,2,3, (4.79) 

7 = 1 



148 



where Tj 6 R 3 is the vector of the internal EAS parameters a' 6 ), and N 1 the Wilson's 
incompatible shape function for the tri-linear brick element 

N 1 « J '.((*0%l) ' / = 1 - 2 - 3 - ( 4 - 8 °) 

The components of T\ are listed in Appendix A. 1 . 

Substituting (4.77) and (4.79) in (4.76), and comparing it with (4.70), the enhancing 
convected base g t , depending on and a^, can be expressed as 

with (a'") - jfaf , a t fc := | (gS-G,) . (4.81) 

The variation and the linearization of the convected basis vector g i and its variation bg { 
with respect to the displacement S e>> and the EAS parameter are given by 

8 9l (d« a< e >) 

A 9l (d (e) ,a (e) ) 

Using (4.82), the variation of components of the strain tensor E takes the form 
SE i:i = sym (to'S*) = s Y m (fai *0j + S 9ok^'9j) + sym (g c ok S^' gj ) , (4.85) 
which can be recast in matrix form as 

{8E tj } = B (S e \ a (e) ) W (e) + 5 (d (e) , a (e) ) <W e) , (4.86) 

where B and B are the strain-displacement matrices associated with Sd^ and 5oS e \ re- 
spectively. 

The linearization of fcEy in (4.85) is 



= 5tf + Sg,, with to = tosb^i + glJT* , (4.82) 
= A^ + Ap^ + ^A^, (4.83) 
= Sg^AT? + bg&r} . (4.84) 



A(<5£^-) = sym[to'A&j+A(to)-^] . 



(4.87) 



149 



where Sg it Ag t , and A (Sg^ are obtained in (4.82)-(4.84). 

Using (4.86) and (4.75), the discrete weak form SU^ iff of the stiffness operator for 
element (e) is then 

SIl% = j {5E lJ } T {S i ^}dV = 6d^ T I B T {S^}dV + 8cx^ T J E? {&>}dV 

= S^ T f% + 6a^f^ St (4-88) 

where /^L and f ^AS are tne internal forces associated with the nodal displacement S e ^ 
and the EAS parameter a^ e \ respectively. 

The linearization of in matrix form then be written as 

V (SU%) • (AdW, A<xV) = 0* T + fc£Aa«) 

+ Sa^ T (k^ u Ad {e) + k^ a Aa^) . (4.89) 

With the constitutive relation {S lj } = C {Eij} for the elastic material, the stiffness 
submatrices JfeW, kQ, and k£> in (4.89) are 

fcS = = / (G£A, + * T ^) dV , (4.90) 



* 2 = iM = S {G T uA + B T CB)dV, (4.91) 



0/^ 

to 

*S2 = fff = [*£f = / (gL$u + B T CB) dV , (4.92) 



(«) 



* 2 = = I {G T aa S a + B T CB)dV, (4.93) 

where the matrices associated with the geometrical stiffness parts in (4.90-4.93) are 

dB(S e \oc^) dB(d (e) ,a< e >) 

Gu " := dd& ' G ** :ss aSw ' 

c - := fldw ' C " :i= &»<■> ' (4 " 94) 



150 



and $„ and S Q are the stress matrices, whose expressions are listed in Appendix A.l. 

The linearized stationarity condition for the mixed functional IT in (4.53) is obtained 
by using (4.89), (4.88), and (4.65), and can be expressed in the following matrix relation 

,(e) _ ,(e) -j 
J ext J stiff I 



K uu 


fe(e) " 


f Ad^ ' 


H 


fc(c) 


fc(e) 
"'act . 







~Jl 



(4.95) 



EAS ) 

recalling that the discretization of SU ext in (4.65), when restricted to element (e) can be 
written as 

1 OT2 = -WM-/S2, ^ (4.96) 

and with /j^ts ' n (4-88), we follow the same procedure for the nonlinear solution as 
in Section 3.3. 

Remark 4.5. It should be noted that in the EAS formulation based on the Green- 
Lagrange Strain E, while the tangent stiffness fej^ in (3.71) contains both the geometric 
part and the material parts, the other three tangent stiffness submatrices fej^, kg£ and fcj£| 
contain only the material part, because while the strain-displacement matrix B (see (3.55) 
and (3.73)) depends on the displacement S e \ the interpolation matrix Q does not. In the 
EAS formulation based on the displacement gradient F, all tangent stiffness submatrices 
as given in (4.90)-(4.93) contain both the geometric part and the metrical part, because 
both strain-displacement matrices B and B (see (4.86) and (4.94)) depend on both the 
displacement and the EAS parameter aS e \ I 

4.3.3. Assumed Natural Strain (ANS) Treatment 

To treat the locking problems due to the transverse shear strains and the transverse 
normal strain, we consider the same approach as in Section 3.3 (i.e., the use of the ANS 
method). To avoid destroying the enhancement of the transverse and normal strains as 
already introduced in the previous section, we need to replace the compatible strain Efj in 
the enhanced strain by the assumed strain E(- Ns , for ij = 13, 23, 33 (i.e., the transverse 
and normal strains only) as follows 

Ekj = En - E% + E* NS , for ij = 13, 23, 33. (4.97) 



151 



where is the modified strain, the compatible strain Efj is given in (4.73), and the as- 
sumed strain E* NS is given in (3.100) and (3.101). 

Since the variation and the increment of the components of the compatible strain E c 

are 



SE^ = S 



\ (rf-flj ~ GfG,)] = sym (fcf.flj) , (4.98) 

A£<= = sym(A^.^ c ) , (4.99) 
respectively, and the linearization of the variation of the compatible strain is 

A = A [sym (Sg^)] = sym (Stf- Agfj , (4.100) 

we use the same procedure in Section 3.3 to obtain the variation 8E(- NS of the assumed 
natural strain and its linearization A \ SE^ NS \. The expressions in (4.85) and (4.87) are 
then replaced by the following 

54- = SEv-dE^+SEf™ , (4.101) 
A(SE ij ) = A(SE ij )-A(6Et J )+A(SEf j NS ) , (4.102) 

for the computation of the residual in (4.88) and for the tangent stiffness submatrices in 
(4.90)-(4.93). 

4.3.4. Simplified Formulation 

In the present EAS formulation based on the deformation gradient F, both the strain- 
displacement matrices B and B are functions of the nodal displacement S e) and element 
EAS parameter a (e) . Such strong coupling makes this EAS formulation more complex than 
the EAS formation based on the Green-Lagrange Strain E, where the strain-displacement 
matrix B (see (3.55)) depends only on S e \ and the interpolation matrix Q depends on 
neither d (e) nor a (e) . Recall that the tangent vector g { can be written as 

9l = g\ (d^) + ~g x (d< e \ a< e >) , with ~ 9i = g% k (rfW) (a«) . (4.103) 



152 



With the Green-Lagrange strain E expressed as 

E=(Et j + E ij )G?<8>G i , (4.104) 

where compatible Green-Lagrange strain component Efj and the enhancing strain com- 
ponent Eij are given in (4.7 1 ). Note that g*t and thus JSy are the functions of d (e) and 

Recall from (3.57) that the enhancing strain in the EAS formulation based on the 
Green-Lagrange Strain E actually depends linearly on a (e) only. 

To simplify the formulation, it is possible to omit the high order term g^g^ in the 
expression for g*j in (4.73), that is, consider the approximation 

+%*«;'. (4-105) 
which, together with (4.82), lead to the following expression for SEij 
SEij = -Sgij * + -6 (gt% + g^gf) 

= sym (Sfrgj + ^Sfc^f '9s) + sym (dfc^f •*}) , (4.106) 

where the two above terms in turn lead to simplified expressions for the strain-displacement 
matrices B and B, respectively. 

We then obtain the following linearization of the variation SE of the strain 

= \A(6g 1J )^^A(8g^ + 1 -A(5(g c r g J +g c r g l )) 
= sym [5g c t - (A^ + A<4^) + Sg^-Ag] 
+sym (SgS-g^Aj* + Sg^A^-g') 

+sym (Ag&j*^ + g&ff'Ag*) , (4.107) 

where the above three terms correspond to the simplified geometric stiffness submatri- 
ces G uu , G ua , and G au associated with [Sd {e \ Ad (e) ), (<5d (e) , AqW), (<W e \ Ad (e) ), 



153 



respectively, while the submatrix G aa associated with (£a (e) , Ac* (e) J (similar to the ex- 
pression in 4.93) vanishes completely as a result of the approximation in (4.105). Readers 
are referred to Appendix A. 1 for details of the above matrices. 

We follow the same procedures described in (4.86), (4.88), and (4.89) to obtain the 
tangent stiffness submatrices that are similar to (4.90)-(4.93) and the internal forces similar 
to (4.88), and then solve the assembled nonlinear problem. Our numerical tests show that 
it is virtually identical for both the full formulation and the simplified formulation. 



The finite element formulation of the present low-order solid-shell element for dy- 
namic analyses of multilayer composite shell structures, presented in the previous sections, 
has been implemented in both Matlab and the Finite Element Analysis Program (FEAP), 
developed by R.L. Taylor (Zienkiewicz and Taylor [1989]), and run on a Compaq Alpha 
workstation with UNIX OSF1 V5.0 910 operating system. The tangent stiffness matrix, 
the dynamic residual force vector, and the consistent mass matrix are evaluated using full 
2x2x2 Gauss integration in each element. A tolerance of 10 -18 on the energy norm 
is employed in the Newton iteration scheme for the convergence. Below we present nu- 
merical examples involving geometrically nonlinear dynamic analyses, with isotropic and 
orthotropic elastic materials. 

To assess the performance of the present solid shell formulation in dynamic analyses 
of multilayer composite shells, we implemented four different second-order implicit inte- 
gration schemes, and compared their performance in relation to our solid-shell formulation. 

For the classical Newmark (trapezoidal) algorithm, the time-integration parameters 
in (4.2)-(4.6) are chosen as follows 



For the energy-momentum conserving algorithm, the time-integration parameters in 
(4.2)-(4.6) are chosen as follows 



4.4. Numerical Examples 




(4.108) 



1 



2 - Poo 

Poo + 1 




a f = 



Poo + 1 ' 



a. 



m 



154 



where p^ is the user-specified spectral radius or high-frequency dissipation coefficient 
(Chung and Hulbert [1993] and Kuhl and Ramm [1999]). The value p^ = 1 corresponds 
to the case of non numerical dissipation, while a smaller spectral radius (p^ < 1) corre- 
sponds to a greater numerical dissipation for the generalized energy-momentum algorithm 
by Kuhl and Ramm [1999]. It is noted that the numerical dissipation can also be introduced 
by shifting the algorithmic stress in (4.9) with a small damping parameter, as done in the 
modified energy-momentum algorithm (Armero and Petocz [1998]). On the other hand, 
both the modified energy-momentum algorithm by Armero and Petocz [1998] and the gen- 
eralized energy-momentum algorithm by Kuhl and Ramm [1999] lose the second-order 
accuracy (Kuhl and Crisfield [1999]). 

By simply replacing the algorithmic stress (4.9) of the average between the config- 
urations at the beginning and at the end of the time step with the stress at the mid-point 
configuration, we recover the mid-point rule from the energy-momentum conserving algo- 
rithm without numerical dissipation. 

The Bossak-a algorithm (Wood et al. [1981]) can be contained as a special case 
of general ized-a method (Chung and Hulbert [1993]), which is second-order accurate, 
and has the controllable numerical dissipation in the higher-frequency modes. The time- 
integration parameters of Bossak-a algorithm are 

2 1 2 1 

a f = l, a m = , p=- (a m ) , 7 = a m - - . (4.1 10) 

Poo i»4 L 

Although the generalized-a method possesses an optimal combination of low numerical 
dissipation in the low-frequency range and high numerical dissipation in the high-frequency 
range (Chung and Hulbert [1993]), the numerical results in Kuhl and Ramm [1999] showed 
that larger numerical dissipation was necessary to obtain a stable integration for nonlinear 
elastodynamics, when compared to the Bossak-a algorithm. 
4.4. 1 . Double Cantilever Elastic Beam under Point Load 

This example concerns the dynamic responses of an linear elastic beam with rect- 
angular cross-section, built-in at both ends, subject to a suddenly applied step load at its 



155 



midspan (see Figure 4.1). This central part of the beam undergoes displacement several 
times its thickness, so that the solution quickly becomes dominated by membrane effects 
which significantly stiffen its response. The purpose of this example is to verify the present 
numerical implementation of implicit dynamic analysis. Five solid-shell elements are used 

Z fP = m £ = 3xl0 7 

Y 



©- 



X 



20.0 



v = 0.3 
p = 2.54 x 10~ 4 



1.0 



T 



0.125 



Figure 4. 1 . Double cantilever elastic beam under point load. Geometry and material prop- 
erty. 

to model one half of the beam, with symmetric conditions (u = 0) applied at the midspan. 
The fixed time step-size is At = 50 x 10 _6 sec, and the Newmark method without numer- 
ical damping is used. The displacement in axis Z at the midspan along time are shown in 
Figure 4.2, in which the results with lumped mass matrix (obtained via (5.82) or (5.83)) is 
close to that from the beam element with cubic interpolation reported in Example 5.2.1 of 
ABAQUS [1995]. For flexural problems such as beams and shells, it is noted that the use 
of consistent mass matrix leads to more accurate results (Cook, Malthus and Plesha [1989, 
p. 375]). The linear and angular momenta along with time are demonstrated in Figure 4.3. 
Table 4.1, which depicts the values of the Euclidean norm of both the residual and the en- 
ergy norm at each iteration, clearly exhibits the quadratic rate of asymptotic convergence 
in the Newton's solution procedure. 

In nonlinear analyses, it is informative to print the energy balance, which allows us 
to assess how much energy has been lost. To this end, the kinetic energy, the strain energy, 
the total energy (kinetic + strain energies), and the work of the external forces are plotted as 
a function of the integration time. Figure 4.4 shows the energy values from both Newmark 
algorithm and the EM algorithm, where the very small energy balance error (total energy 
versus external work) indicates the Newmark algorithm gives the satisfactory solutions. 



156 




Figure 4.2. Double cantilever elastic beam under point load. Dynamic response at midspan 
of beam. 

Table 4.1. Double cantilever elastic beam under point load: Convergence results (residual 
norm, energy norm). 



Iter. 


Time step 1 (t=5E-5 sec) 


step 40 (t=2E-3 sec) 


step 100 (t=5E-3 sec) 





4.525£+02,8.236£+00 


2.009£+03, 1.804E+02 


2.442£+03, 2.476£+02 


1 


4.479£+03,3.851£-02 


2.555£+04, 1.327E+00 


2.419£+04, 1.179E+00 


2 


3.558£-01,4.407£-10 


1.551.0+01, 3.053£-05 


1.029.E+01, 1.515£-06 


3 


1.125£-07,1.731£-22 


3.917£-02,2.737£-12 


4.757£-04,5.898£-16 


4 




4.833£-09,5.385£-26 


2.218£-08, 1.536E-24 




Figure 4.3. Double cantilever elastic beam under point load. Three components of linear 
momentum (left) and of angular momentum (right), using Newmark algorithm. 



157 




Figure 4.4. Double cantilever elastic beam under point load. Energy conservation using 
the Newmark algorithm (left) and the EM algorithm (right). 



4.4.2. Pinched Cylindrical Multilayer Shell 

Here we simulate the large deformation of a pinched cylindrical shell with ply drop- 
off by using the classical Newmark algorithm. The geometry properties, material parame- 
ters, finite-element mesh and loading conditions for one-eighth are given in Figure 4.5. The 
cylindrical shell is subjected to two opposite forces acting at the mid-section and on the 
outer surface of the shell. The initial conditions for displacements and velocities at t = 
are set to zero. One-eighth of the cylinder is discretized with a FE mesh 32 x (32 + 8) 
solid-shell elements, with appropriate boundary conditions at the plane of symmetry. The 
cylinder is pinched with three different rates of loading as shown in Figure 4.5. Smaller 
time-step size is used for larger rate of loading, that is, 0.04 sec for Rate 1, 0.02 sec for 
Rate 2, and 0.01 sec for Rate 3. The deformed shape for static analysis is shown on the 
left of Figure 4.7, while the deformed shapes for dynamic analyses with the above men- 
tioned rates of loading are shown in the right of Figure 4.7 and Figure 4.8. On average, 
it took roughly five iterations to convergence for each time step in the dynamic analyses. 
Figure 4.6 displays the relationship between the magnitude of the pinching forces and the 
displacement at the point of application of a pinching force. The dynamic analyses clearly 
lead to patterns of deformation that are more complex than that obtained with static anal- 
ysis. Both the inertia (mass) and the loading rates have important influence on the final 
deformed shapes, in which buckling modes along both the circumferential direction and 



158 



the longitudinal direction can be clearly observed. At a higher rate of loading, larger mag- 
nitude of the pinching force is required for the same displacement; the overall resulting 
deformed shape is also more severe. 




Figure 4.5. Pinched cylindrical shell with ply drop-off: Geometry, material, loadings, 0° 
along circumferential direction. 




50 100 150 200 250 300 



Displacement 

Figure 4.6. Pinched cylindrical shell with ply drop-off: Pinching force amplitude versus 
displacement under pinching force. 



159 




Figure 4.7. Pinched cylindrical shell with ply drop-off: Deformed shape for static case 
(left), deformed shape at t = 2.64 for Rate 1 , time-step size 0.04 sec (right). 




Figure 4.8. Pinched cylindrical shell with ply drop-off: Deformed shape at t = 1 .98 for Rate 
2, time-step size 0.02 sec (left), deformed shape at t = 1.5 for Rate 3, time-step size 0.01 
sec (right). 

4.4.3. Free-Flying Single-Layer Plate 

The same example as in Kuhl and Ramm [1999] is used here to compare the perfor- 
mance of the present solid-shell formulation to that of the eight-node shell element with 
reduced integration. The geometry, the loading configuration and the time history loading 
amplitude are described in Figure 4.9. In this example, we use thirty solid-shell elements, 
with a time-step size of At — 50 x 10" 6 sec and a total simulation time of t =* O.lsec. The 
effect of gravity force is not considered. 



160 



The material properties are 

E = 206.GPa, v = 0., p = 7800.i^/m 3 , (4.1 1 1) 

where E, u, and p are the Young's modulus, the Poisson's ratio, and the mass density, 
respectively. 

Snap shots of the plate undergoing large overall motion and large deformation are 
taken every 4 x 10~ 3 sec from the simulation by using the energy-momentum (EM) con- 
serving algorithm, and are displayed in Figure 4.11. Furthermore, the conservation of 
linear and angular momenta is demonstrated in Figure 4.12. On the left of Figure 4.16, the 
kinetic energy, the strain energy, the total energy (kinetic + strain energies), and the work 
of the external forces are plotted as a function of the integration time. Unlike the slightly 
increasing total energy caused by reduced integration technique in Kuhl and Ramm [1999], 
the total energy in the present solid-shell formulation is exactly conserved. The classical 
Newmark method and the mid-point rule lead to a loss of stability at an early time stage of 
the integration, as indicated by the dramatic increase of the total energy (Figure 4.17). The 
mid-point rule has a stability that is somewhat worse than that of the Newmark algorithm 
in this example. On the other hand, the mid-point rule conserves the angular momentum, 
while the Newmark algorithm conserves only the linear momentum (Figure 4.13 and Fig- 
ure 4.14). With a numerical dissipation set at p^ = 0.975, the Bossak-a method yields a 
stable integration in the whole time range (the right of Figure 4. 16), and conserves both the 
linear momentum and the angular momentum. The variation of the number of iterations per 
time-step as the integration progressed indicates a trend of increasing number of iterations 
and a potential for lack of convergence at some future station in the EM algorithm (left of 
Figure 4. 1 8). On the other hand, the Bossak-a algorithm possesses a stable number of iter- 
ations and thus a better rate of convergence at each time step (right of Figure 4. 1 8). While 
the difference in the displacements obtained from the EM algorithm and from the Bossak- 
a algorithm is negligible (left of Figure 4.19), the difference in the total energy is about 
18% at t = O.lsec (Figure 4.16). In parallel, the difference in the velocity (right hand side 



161 

of Figure 4.19) is small, but the difference in the acceleration (Figure 4.20) is large. The 
magnitude of the acceleration obtained from the EM algorithm increases rapidly, while the 
one obtained from the Bossak-a algorithm hovers at around the level of 10 6 . The relative 
higher magnitude of acceleration suggests a higher level of "noise" in the high-frequency 
range of the response obtained from the EM algorithm, compared to the Bossak-a algo- 
rithm, whose parameter a m in (4.1 10) has a direct influence on the acceleration update as 
expressed in (4.3). 



0.04m 




Figure 4.9. Free-flying single-layer plate: Geometry, loading distribution, and loading 
amplitude. 




Figure 4.10. Free-flying single-layer plate: Initial undeformed configuration at t — 0. 
4.4.4. Free-Flying Multilayer Plate with Ply Drop-offs 

This example establishes the capability and performance of the present solid-shell 
in modeling multilayer plates/shells with ply drop-offs. Comparison of the stability and 
accuracy of the four different time-integration algorithms is provided for this example. The 
geometry and the mesh of a three-layer plate with ply drop-offs is shown in Figure 4.21. 



162 




Figure 4.11. Free-flying single-layer plate: Perspective view. 

4 5 
4 

3.6 - 



1 
0.5 


0.01 0.02 0.OS 0.04 0.0S 0.06 0.07 0.0» 0.0» 0.1 0.01 02 03 0.04 05 06 07 06 00 1 

l" ) Tm [sec) 

Figure 4.12. Free-flying single-layer plate: Three components of linear momentum (left) 
and of angular momentum (right), using EM algorithm. 

The length, width, and layer thickness of the plate are the same as those of the single- 
layer plate in Subsection 4.4.3 (i.e., L m 0.3, W = 0.06, and {e) h = 0.001). The plate 
is divided into three equal parts and two ply drop-offs along its length, with each part 
having a length of 0.1. We use the same material properties as listed in (4.1 1 1) so that the 
three-layer plate with ply drop-offs here has the same weight as the the single-layer plate 



• -0.15 

| -0.2 
I 
-0.26 



0.01 0.02 0.03 0.04 0.05 0.00 0.07 0.08 0.0» 0.1 " 0.01 0.02 03 0.04 0.05 06 07 06 0« 1 

Figure 4.13. Free-flying single-layer plate: Linear momentum (left) and angular momen- 
tum (right) using the Newmark algorithm. 



0.01 0.02 0.03 0.04 0.06 0.06 
Tims (sac) 



-0.1 

| -0.15 

t -0.2 
I 
I 
% -0.25 



— 1 1 |— 



0.07 0.08 0.09 0.1 01 0.02 0.03 DM O.OS 006 0.07 0.06 0.09 0.1 

Time (sec) 



Figure 4.14. Free-flying single-layer plate: Linear momentum (left) and angular momen- 
tum (right) using the mid-point rule. 



- Lx 

- - Ly 



0.01 0.02 0.03 0.04 0.05 06 0.07 0.08 09 0. 

Tim* : sec i 



-0,1 
• -0 15 



■§ -0.2 
I 

-0.25 



0.01 0.02 0.03 0.04 0.05 0.06 0.07 08 09 1 

Tim* (s*c) 



Figure 4.15. Free-flying single-layer plate: Linear momentum (left) and angular 
turn (right) using the Bossak-a algorithm. 



momen- 



in Subsection 4.4.3. The load distribution and the time history of the loading amplitude 
are also the same as Figure 4.9. A total of sixty solid-shell elements are used to model the 



164 



!! 'f. 



i'i! 
i I! 



s. 



i! i! 



i ' » V 



is 5 

i"; h ?, i\ 

ti i» ! * 



s 1 



5 2 

(I W 

mm 



— Kinetic 
Strain 

Kinetic+Strain 

External Work 



* i i 



5 1 



MAI 



i m 



., "' •} „ A 5 



i K .n n il "i H « A A- M i 11 ■ i 
* n * f V -if 1/ y y U i is * 1 



- Kinetic 
Strain 

Kinetic* Si rain 

— External Work 



0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0. 1 
Time (sac) 



0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 
Tims (sac) 



Figure 4.16. Free-flying single-layer plate: Energy conservation using the EM algorithm 
(left) and the Bossak-a algorithm (right). 



i--vs 



Kinetic 

Strain 
• Kinetlc+Straln 
- External Work 



: i : ■ ! ■ i ii ; 



Kinetic 

Strain 

Kinetic+Strain 

External Work 



0.01 O.02 0.03 0.04 0.05 0.06 0.07 0.0B 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

TVna (aae) Time (aac) 

Figure 4.17. Free-flying single-layer plate: Energy conservation using the Newmark algo- 
rithm (left) and the mid-point rule (right). 




Figure 4.18. Free-flying single-layer plate: Number of iterations to convergence at each 
time step using the EM algorithm (left) and Bossak-a algorithm (right). 

three-layer plate with ply drop-offs. The same time-step size and time range are used as in 
Subsection 4.4.3. 




Tim* (mc) Tim* (sac) 

Figure 4.19. Free-flying single-layer plate: Difference in displacements (left) and in veloc- 
ities (right) as obtained from the EM algorithm and the Bossak-a algorithm. 



1 1 1 1 1 1 1 1 










EM 

Bossak 






i i i i i i i 


i 


i 



0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

Time (sec) 

Figure 4.20. Free-flying single-layer plate: Difference in accelerations as obtained from 
EM algorithm and Bossak-a algorithm. 

Snap shots of the deformed shapes taken at every 4 x 10 -3 sec time interval from 
the computations using the Bossak-a algorithm are presented in Figure 4.22. The energy 
distribution for this example is given in Figure 4.28, which when compared to Figure 4. 16 
reveals a smaller level of total energy in the present example of the three-layer plate in 



166 

relative to the single-layer plate in Subsection 4.4.3. The deformed shapes of the single- 
layer plate and of the three-layer plates are shown in Figure 4.23, where it can be seen 
that the three-layer plate has a more flexible (thinner) end and a more rigid (thicker) end 
when compared to the single-layer plate. Further, when comparing the initial undeformed 
configuration in Figure 4.21 to the deformed configuration on the right of Figure 4.23, we 
can see that the thick end of the plate has moved from left to right as a result of the large 
overall rotation of the plate. Both Figure 4.21 and Figure 4.23 were plotted using the same 
perspective point of view. 

In this example, the energy-momentum conserving algorithm could not carry on the 
integration beyond the time t = 56 x 10" 3 sec due to lack of convergence (right of Fig- 
ure 4.28) . On the other hand, the Bossak-a algorithm with the numerical dissipation 
Poo = 0.975 provides a stable integration in the whole time range, and conserves the lin- 
ear and angular momenta (Figure 4.27). With the EM algorithm, the linear and angular 
momenta (Figure 4.24) together with the energy (left of Figure 4.28) are conserved for 
the present solid-shell element up to the time t = 56 x 10 _3 sec. The difference between 
the displacements obtained with the EM algorithm and with the Bossak-a algorithm up 
to t = 56 x 10~ 3 sec is negligible (left of Figure 4.31) ; the difference in the velocity is 
negligible up to about t — 20 x 10~ 3 sec, and then increases to a noticeable level until 
t = 56 x 10 _3 sec (right of Figure 4.31). The total energy with the Bossak-a algorithm 
exhibits a loss of about 24%. Again the difference in the acceleration is large (Figure 4.32). 
The increasing number of iterations to convergence in each time step when using the EM al- 
gorithm (Figure 4.30) eventually leads to a numerical failure at time t = 56 x 10 _3 sec. The 
reason behind this increase in the number of iterations and the eventual numerical failure 
is the increasingly active high-frequency response in the numerical solution as manifested 
in the rapid increase of the acceleration (Figure 4.32). Similar to the single-layer plate in 
Subsection 4.4.3. The Newmark algorithm and the mid-point rule lead to a loss of stability 
at an early stage in the solution process, as shown in Figure 4.29. On the other hand, the 
loss of stability of the mid-point rule occurred earlier compared to the Newmark algorithm, 



167 

even though the mid-point rule conserves both the linear and angular momenta, while the 
Newmark algorithm conserves only the linear momentum (Figure 4.25 and Figure 4.26). 

Among the recent energy-momentum conserving algorithms with numerical dissi- 
pation, the algorithm by Kuhl and Ramm [1999] is shown to be more robust than the 
algorithm by Armero and Petocz [1998]. The level of acceleration in the Kuhl and Ramm 
[1999] algorithm plateaued out after an initial stage of increase in magnitude, and stayed 
that way until the end of the simulation time range (left of Figure 4.36). The acceleration 
in the Armero and Petocz [1998] algorithm kept increasing until a lack of convergence that 
prematurely halted the solution process in the same manner as encountered with the orig- 
inal EM algorithm by Simo, Tarnow and Wong [1992] (right of Figure 4.36). Figure 4.37 
displays the number of iterations to converge in the Kuhl and Ramm [1999] algorithm and 
in the Armero and Petocz [1998] algorithm. The difference in displacements obtained from 
both EM algorithms with numerical dissipation and the Bossak-a algorithm is negligible 
(Figure 4.34), and the difference in the velocity (Figure 4.35) is small. Nevertheless, unlike 
the Bossak-a algorithm, both EM algorithms with numerical dissipation lose the desired 
second-order accuracy (Kuhl and Crisfield [1999]). 




Figure 4.21. Free-flying three-layer plate with ply drop-offs: Initial undeformed configura- 
tion at t = 0. 

From the results of above numerical examples, it is clear that the Newmark (trape- 
zoidal) algorithm and the mid-point rule do not guarantee a robust time integration in non- 



168 




Figure 4.23. Free-flying single-layer plate (left) and three-layer plate (right) with ply drop- 
offs: Deformed shapes at t = 16 x 10 -3 sec. . 



linear elastic dynamics due to their rapidly increasing energy in the integration process. By 
algorithmically enforcing the conservation of the total energy within each time step, the 
energy-momentum algorithm gains a more robust stability behavior compared to the New- 
mark algorithm and the mid-point rule. Yet, the EM algorithm remains unstable due to a 
continuous growth in the acceleration; such accelerated growth would eventually terminate 



169 




Figure 4.24. Free-flying three-layer plate with ply drop-offs: Linear momentum and angu- 
lar momentum using the EM algorithm. 




Figure 4.25. Free-flying three-layer plate with ply drop-offs: Linear momentum and angu- 
lar momentum using the Newmark algorithm. 




Figure 4.26. Free-flying three-layer plate with ply drop-offs: Linear momentum and angu- 
lar momentum using the mid-point rule. 

the integration process due to a lack of convergence within a finite time length. Quenching 
the acceleration growth in the EM algorithm by introducing numerical damping in the high- 



170 




-0.05 ] \ 



■- Jx 
- - Jy 



01 0.02 0.03 0.04 0.05 0.06 0.07 O.OB 0.09 0.1 0.01 0.02 0.03 0.04 0.05 O.OG 0.07 0.08 0.09 0.1 

Time (mc) Time (sec) 

Figure 4.27. Free-flying three-layer plate with ply drop-offs: Linear momentum and angu- 
lar momentum using the Bossak-a algorithm with = 0.975. 



r, ' *i 



Kinetic 
Strain 

- Kinetic+Sttaln 

- External Work 



. n my ^r^ 1 



jV /\ k, \ 



' V * S VV''\A i S-y;-, 



- Kinetic 
Strain 

- - Kinetic+Slrajn 

— EmarnaiWork 



i : i 



0.01 0.02 0.03 0.04 0.0S 0.06 0.07 0.06 0.0ft 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.06 O.OB 0.1 

Time (sec) Time (sec) 

Figure 4.28. Free-flying three-layer plate with ply drop-offs: Energy conservation and 
divergence using the EM algorithm (left); energy loss and continued integration using the 
Bossak-a algorithm with p^ = 0.975 (right). 




■-■ Kinetic 
Strain 
— Kinetic +St rain 
External Work 



0.01 0.02 0.03 0.04 0.06 0.06 0.07 0.08 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

Time (»ec) Time (aec) 

Figure 4.29. Free-flying three-layer plate with ply drop-offs: Energy balance and diver- 
gence using the Newmark algorithm (left) and the mid-point rule (right). 




I 1 1 1 1 1 1 1 1 1 1 o 1 1 1 ' 1 ' 1 1 1 ' 1 

0.01 0.02 0.03 0.04 0.0S 0.06 0.07 0.08 0.09 0. 1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.06 0.09 t 

T)m* (uc) Tim* (sac) 

Figure 4.30. Free-flying three-layer plate with ply drop-offs: Number of iterations till 
convergence in each time step for the EM algorithm (left) and for the Bossak-a algorithm 
with poo = 0.975 (right). 




0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.0B 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.06 0.09 0.1 



Figure 4.31. Free-flying three-layer plate with ply drop-offs: Difference in displacements 
(left) and in velocities (right) as obtained from the EM algorithm and the Bossak-a algo- 
rithm with poo = 0.975. 

frequency range may or may not prolong the termination of the integration process, while 
paying the price of losing the desired second-order accuracy. All EM algorithms, with or 
without numerical damping, lead to non-symmetric tangent stiffness matrices. By contrast, 
the Bossak-a algorithm with an appropriate amount of numerical dissipation provides a 
stable, and second-order accurate integration process that yields practically the same dis- 
placements as obtained in all other algorithms (Newmark, mid-point rule, EM algorithm 
with or without numerical damping) before the failure. We note on the other hand that 
there is a smaller loss of total energy in the algorithm of Kuhl and Ramm [1999] (left of 
Figure 4.33) when compared to the Bossak-a algorithm (right of Figure 4.28). 



172 



10 




Figure 4.32. Free-flying three-layer plate with ply drop-offs: Difference in accelerations 
obtained from the EM algorithm and Bossak-a algorithm with = 0.975. 




0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.01 0.02 0.03 0.04 05 06 0.07 0.08 0.09 0.1 

TV"* (»•«) Hm« (««c) 



Figure 4.33. Free-flying three-layer plate with ply drop-offs: Energy balance using the EM 
algorithm with = 0.985 as in Kuhl and Ramm [1999] (left) and with f = 0.0001 as in 
Armero and Petocz [1998] (right). 



173 




Figure 4.34. Free-flying three-layer plate with ply drop-offs: Difference in displacements 
using the EM algorithm with Poo = 0.985 as in Kuhl and Ramm [1999] (left) and with 
£ = 0.0001 as in Armero and Petocz [1998] (right), compared to the Bossak-a algorithm 
with poo = 0.975. 




Time (uc) TJma (sec) 

Figure 4.35. Free-flying three-layer plate with ply drop-offs: Difference in velocities using 
the EM algorithm with p^ = 0.985 as in Kuhl and Ramm [1999] (left) and with f = 
0.0001 as in Armero and Petocz [1998] (right), compared to the Bossak-o algorithm with 
Poo = 0.975. 



174 




1 1 1 1 1 1 — 


1 EM 

| - - Bossak | 


i i i i i i 





0.01 0.02 0.03 0.04 0.0S 0.06 0.07 0.0B 0.09 0.1 01 0.02 0.03 0.04 0.05 0.08 0.07 0.0B 0.09 0.1 

Time (ssc) Time (sec) 

Figure 4.36. Free-flying three-layer plate with ply drop-offs: Difference in accelerations 
using the EM algorithm with = 0.985 as in Kuhl and Ramm [1999] (left) and with 
£ = 0.0001 as in Armero and Petocz [1998] (right), compared to the Bossak-a algorithm 
with poo = 0.975. 




0.08 0.09 0.1 



Time (sac) Time (sec) 

Figure 4.37. Free-flying three-layer plate with ply drop-offs: Number of iterations till 
convergence in each time step using the EM algorithm with p^ = 0.985 as in Kuhl and 
Ramm [1999] (left) and with f = 0.0001 as in Armero and Petocz [1998] (right). 



CHAPTER 5 

EFFICIENT AND ACCURATE MULTILAYER SOLID-SHELL 
ELEMENT: NONLINEAR MATERIALS AT FINITE STRAIN 

5.1. Introduction 

The analysis of general shell structures have been of interest for several decades. 
There is a continuing challenge to develop reliable, accurate and efficient low-order shell 
elements, especially for analyses of shell structures with arbitrary geometries, loadings, 
boundary conditions and nonlinear materials. 

Because of the high cost of 3-D continuum elements, shell structures are mainly mod- 
eled by shell elements based on either the degenerated shell concept or the classical stress- 
resultant shell theory. Both formulations are based on the common kinematic assumptions 
of inextensibility in the thickness direction and the zero-transverse-normal-stress condition. 
Although these approximations led to good results in most cases, several difficulties and 
appreciable errors could arise. Since the zero-transverse-normal-stress condition must be 
imposed, the implementation of 3-D material models proves to be a difficult task, and the 
complexity of the algorithmic treatment is increased. For example, even for the simplest 
von-Mises elastoplastic model, the stress-resultant constitutive models is rather complex 
(Simo and Kennedy [1992]). Moreover, a proper description of 2-D constitutive equations 
at finite strain remains a question (Schieck, Pietraszkiewicz and Stumpf [1992]). On the 
other hand, in many applications involving (i) the localized effects due to surface load- 
ings, (ii) the contact interaction of different shell structures, or (iii) the delamination of 
multilayer shells, it is important to include the transverse normal-stress and the associated 
thickness change to obtain a better accuracy (Cho, Yang and Chung [2002], Fox [2000]). 
Furthermore, when both shell elements and solid elements are used in one FE model such 



176 



as folded shell structures, 12 additional transition elements (e.g., Liao, Reddy and Engelstad 
[1988], Cofer and Will [1992]) or multipoint constraints (e.g., MPC in ABAQUS [2001]) 
are needed to connect rotational dofs and displacement dofs. 



solid-shell elements 




weld zone modeled by solid element 





Figure 5.1. Discretization at shell junction: Combination of solid elements and solid-shell 
elements. 

The proposed solid-shell element formulation overcomes the above mentioned diffi- 
culties, and improves the computational accuracy for wider shell applications. The kine- 
matic description of the present element consists of only displacement dofs at the top and 
bottom surfaces of the shell. Complex finite rotation updates such as those found in stress- 
resultant shell elements (Vu-Quoc, Deng and Tan [2000]). The present formulation also 
provides a natural way to connect to regular solid elements (see Figure 5.1) without the 
need for transition elements or submodeling technique as in ABAQUS [2001], in which 
the modeling processes are laborious and error prone. For bending-dominated problem of 
homogeneous shells, in contrast to the use of 3-D solid elements where a large number of 
layers of elements must be used in the thickness direction together with a dense mesh on 
the shell surface (thus leading to ill-conditioned stiffness matrices), only a single solid shell 
elements across the shell thickness, together with a much coarser mesh of solid shell ele- 
ments on the shell surface, are sufficient to provide accurate results. Unlike stress-resultant 
shell formulations, 3-D nonlinear complex material models can be used directly in a solid- 
shell, without further treatments which add complexity of the model, such as accounting for 

12 It is not possible for stress-resultant shell elements to describe the detailed strain and stress distributions 
at shell intersections (Chroscielewski, Makowski and Stumpf [1997]). 



177 



the plane-stress constraint condition. Moreover, the strain-driven character of the formu- 
lation further simplifies the implementation of nonlinear constitutive models, in contrast 
to the hybrid finite-element formulations derived from the Hellinger-Reissner functional 
(Simo et al. [1989]). Because of the use of the covariant Green-Lagrange strain tensor 
without ignoring the higher order terms, all kinematic quantities such as displacements and 
the corresponding strains can be finite, and the update procedure can be proceeded in an 
exact manner, without approximations. It is noted that the quadratic terms in the strains 
become important in the analysis of relatively thick shells, involving strong curvatures or 
presenting large strains together with bending deformations (Buchter et al. [1994]). 

Displacement-based solid elements are known to have poor performance in bending- 
dominated situation and/or with incompressible materials, such as in thin shell analysis 
with elastoplastic material. To obtain the same performance in bending as with stress- 
resultant shell formulations with plane-stress assumption (e.g., Vu-Quoc, Deng and Tan 
[2000]), the Enhanced Assumed Strain (EAS) method and the Assumed Natural Strain 
(ANS) method are employed in the present solid-shell formulation. 

As demonstrated in Chapter 3, many solid-shell formulations unfortunately do not 
pass the out-of-plane bending patch test (e.g., Miehe [1998&], Klinkel et al. [1999]). To 
avoid the Poisson-thickness locking problem (Zienkiewicz and Taylor [1991, p. 161], Bischoff 
and Ramm [1997]), the transverse normal strain must have at least a linear distribution 
across the shell thickness. In the present formulation, an optimal number of EAS parame- 
ters is established to enhance both the transverse normal strain (linear distribution in thick- 
ness) and the in-plane strains. Moreover, the present solid shell formulation passes both 
the membrane patch test and the out-of-plane bending patch test, and thus with minimum 
computational effort. 

ANS interpolation has been a most successful tool to tackle the shear-locking effect 
in the 4-node displacement-based shell elements, even for initially distorted meshes (Mac- 
Neal [1978], Hughes and Tezduyar [1981], Dvorkin and Bathe [1984], Parisch [1995]), as 
compared to the (selectively) reduced integration. To treat transverse shear locking, we ap- 



178 



ply an ANS interpolation on the compatible transverse shear strains. In the case of curved 
structure with geometric nonlinearity, to treat the curvature-thickness locking (Bischoff and 
Ramm [1997]), which is also called the trapezoidal locking (Sze and Yao [2000]), we apply 
an ANS interpolation on the compatible transverse normal strain, as proposed by Betsch 
and Stein [1995]. 

In addition to the above features, our new contributions in this chapter are specifically 
listed below: 

1) Demonstrate that the proposed optimal seven EAS parameters (three for the transverse 
normal strain to treat the Poisson-thickness locking, and four for the membrane strains to 
treat the in-plane bending locking) are sufficient to avoid locking problems with incom- 
pressible materials, in addition to passing the membrane and out-of-plane bending patch 
tests. 

2) Justify the use of the present element with various nonlinear materials in problems in- 
volving multilayer composite shells, including junctions with regular solid elements and 
contact/impact. 

3) Show that dynamic analyses can be carried out using either the consistent mass matrix 
or lumped mass matrix (in explicit integration 13 ) without spurious modes (Belytschko, Lin 
and Tsay [1984] , Zeng and Combescure [1998]). Recall that the consistent mass matrix 
for multilayer stress resultant shells is complex and configuration dependent (Vu-Quoc et 
al. [2001]). 

Two nonlinear 3-D material models at finite strain have been implemented in our 

solid-shell formulation, and the simulation results reported in the present chapter. For 

the (compressible) Mooney-Rivlin Model, the approach of the incompressibility limit is 

tackled by the use of the penalty method or the augmented Lagrangian method. Other 

forms of hyperelastic constitutive models such as the Ogden-type model (Ogden [1984]) 

are amenable to be implemented in the present element formulation. On the hyperelasto- 

13 It is noted that the rotational dofs in traditional shell formulations correspond to high frequency modes, 
which will drive the stable time-increments to very small size, thus requiring an artificial scaling of the 
rotational masses (Hughes [1987, p.564]). 



179 



plastic model, the current implementation possesses the following advantages: (1) the re- 
turn mapping algorithm of infinitesimal plasticity can be carried over to the present finite 
deformation context without any modification, and with a simplification of the compu- 
tational procedure: The closest-point projection algorithm is now formulated in princi- 
pal stretches. In particular, the algorithmic elastoplastic moduli tensor is symmetric, and 
the incompressibility is automatically ensured; (2) With the elastic response emanating 
from the hyperelastic form of the free-energy function, the elastic predictor in the return- 
mapping algorithm is exact, and computed without resorting to the use of incrementally 
objective algorithms (Simo and Hughes [1998, (p.276)]) as for hypoelastic models; The 
present implementation (3) employs the consistent tangent moduli tensor, instead of the 
continuum elastoplastic moduli tensor, thus achieving quadratic rate of convergence in the 
Newton iterative procedure (Simo [1988Z?]), and (4) finite-strain elastoplasticity based on 
the Cauchy-Green strain tensor, thus avoiding the computation of the deformation gradient 
F via the costly polar decomposition. 

Even though there is no consensus on the necessity of passing the out-of-plane bend- 
ing patch test for convergence, we demonstrate that elements (with insufficient EAS pa- 
rameters) that did not pass the out-of-plane bending patch test perform poorly in problems 
involving nonlinear material behavior and large deformation, as opposed to the present 
formulation, which provides accurate results. 14 

The outline of the present chapter is as follows. We discuss the implementation of the 

Mooney-Rivlin material model and the hyperelastoplastic material model, in Section 5.2. 

The explicit integration method for solid-shell elements is present in Section 5.3. Several 

numerical examples that illustrate the performance of the present formulation involving 

large deformation, implicit and explicit dynamic analyses, together with a comparison of 

the computed results to those obtained from other shell formulations and from a meshless 

method, are presented in Section 5.4. 

14 There is on the other hand a consensus that passing the membrane patch test is necessary for 
convergence. 



180 



5.2. Nonlinear Material Law 

A major advantage of the present solid-shell element is that all algorithms concerning 
the 3-D nonlinear material models can be implemented without any modification. For non- 
linear materials, the second Piola-Kirchhoff stress tensor S in (3.16) and the fourth-order 
consistent tangent moduli tensor C in (3.70) are crucial for the numerical solution of initial- 
boundary-value problems with the quadratically convergent Newton-Raphson scheme. 
5.2.1. The Mooney-Rivlin Material Models 

There are two possible ways to define the stored energy function W s for the modified 
Mooney-Rivlin material, in which both satisfy the zero stress condition S = at the initial 
configuration (i.e., E = 0). 

According to Fried and Johnson [1988], the stored energy function W a of a modified 
Mooney-Rivlin material is expressed as follows 

W s = d (/, - 3/ 3 1/a ) + C 2 (l 2 - 3/ 3 2/3 ) + ^ (ln/ 3 ) 2 , (5.1) 

where the invariants I\, I 2 , and 7 3 are expressed in terms of the right Cauchy-Green tensor 
C 

C = F T F = 2E + 1, (5.2) 

as follows 

h := trace (C) , I 2 := \ [l x 2 - trace (C 2 )] , J 3 := det (C) , (5.3) 

where C\, C 2 are material constants, and A the penalty parameter for incompressibility. 
From (5.3), the following derivatives are obtained 

where 1 represents the second-order identity tensor, and I the fourth-order identity tensor, 
both expressed in the convected basis {Gi} as follows 

1 = G ij Gi ft Gj = G Ij G i ® G ] , (5.5) 
1=1 (p ik G jl + G ll G ]k ) G t 9 Gj ® G k ® G l , (5.6) 



181 

The second Piola-Kirchhoff stress tensor S is the derivative of the stored energy function 
W s with respect to the strain tensor E. Using (5.1) and (5.4), it follows that 



(Ci + C 2 h) 1-C 2 C- (d/ 3 1/3 + 2C 2 / 3 2/3 - Aln/ 3 ) C- 1 ] , (5.7) 



where from (3.1 1) and (5.2) we obtain the following expressions for C and its inverse C 1 
as 

C = F T F = g^G* ® G j , C- 1 = g ij Gi <g> G j , (5.8) 

respectively, with g {j and g lj computed as shown in (3.9) and an equivalent equation. 

The fourth-order material moduli tensor C is the second derivative of the stored en- 
ergy function W s with respect to the strain tensor E, that is 

d 2 W< dS 



C = 



OEdE dE 

dC~ l 



4C 2 1 01- 4C 2 I + 4 (Aln/ 3 - / 3 1/3 d - 2/ 3 2/3 C 2 ) 



dC 



+ 4 (A - i/3 1/3 C, - ll 2 3 /3 C 2 ) C 1 ® C" 1 , (5.9) 
dC~ x 

where the term can be shown to take the form as follows 15 

dC~ l 1 

-QQ- = -g (^ ifc ^' + <?V fc ) ® C?^ <8>G k ®Gi . (5.10) 

Another choice of the energy function W s for the Mooney-Rivlin material is given 
below (Bathe [1996, p.593]): 

W s = d (/, - 3) + C 2 (J a - 3) - {Ci + 2C 2 ) ln/ 3 + ^ (m/ 3 ) 2 , (5.1 1) 

where the parameters C\, C 2 , A have the same meaning as in (5.1). 16 

15 By differentiating the identity g^g jq = St with respect to g kl , and by recognizing that = 
j 9g kt 

2 (9jrg g s + 9js9qr), one arrives at (5.10) after postmultiplying the resulting equation with g qp and reusing 

the identity g jq g vp = 5? once more. 

16 Both (5.1) and (5.1 1) are modified forms of the original Mooney-Rivlin model (Truesdell and Noll 
[1992, p.350]), which corresponds to the first two terms in (5.1 1). 



182 



The second Piola-Kirchhoff stress tensor S corresponding to W s in (5.1 1) is then 
given by 

_ dW, 



3 =2 



(Cj + dh) 1 - C 2 C - (d + 2C 2 - Aln/ 3 ) C- 1 , (5.12) 



dE 

while the material moduli tensor C takes the form 



C = ~ = 4C 2 1 ® 1 - 4C 2 I - 4 (Ci + 2C 2 - Aln/ 3 ) -xpr + 4AC -1 <g> C" 1 . (5.13) 

Our numerical experiments show that both models in (5.1) and (5.11) lead to the 
same results if the same value of penalty parameter A is used. 

Remark 5.1. To satisfy the incompressibility constraint (i.e., h = 1), the penalty 
parameter A must be large enough so that the error on incompressibility is negligible (i.e., 
I 3 is approximately equal to 1). On the other hand, the penalty parameter A cannot be 
so large that numerical ill-conditioning occurs, and makes Newton's method difficult to 
converge. Alternatively, the augmented Lagrangian method is designed to avoid the char- 
acteristic ill-conditioning of the above penalty method, and enforces the incompressibility 
accurately (Simo and Taylor [1991]), but destroys the desired quadratic rate of asymptotic 
convergence of Newton's method (See Section 5.4 for some examples). I 

5.2.2. The Hyperelastoplastic Model 

Up to the beginning of the 1980s, computational methods for finite-strain elastoplas- 
ticity typically relied on hypoelastic extensions of the classical infinitesimal model (Simo 
and Hughes [1998]), and hence not suitable for applications involving large elastic strains 
(e.g., metal forging). In the last 15 years, computational approaches based on the multi- 
plicative decomposition have received considerable attention in the literature. Simo and 
Ortiz [1985] and Simo [1988a] proposed a computational approach entirely based on the 
multiplicative decomposition, and pointed out the role of the intermediate configuration 
in a definition of the trial state via the hyperelastic stress-strain relations. Subsequently, 



183 



Eterovich and Bathe [1990] and Weber and Anand [1990] used the multiplicative decompo- 
sition in conjunction with a logarithmic stored energy function and an exponential approx- 
imation to the flow rule cast in terms of the full plastic deformation gradient. Based on the 
above multiplicative decomposition, Simo [1992] showed that the closest-point-projection 
algorithm of infinitesimal plasticity could be carried over to the finite-deformation context 
without modification. A computational treatment of plasticity is to interpret the local evo- 
lution equations describing the plastic flow in the framework of the principle of maximum 
dissipation, which lead to the return mapping algorithm with an operator split procedure. 
Within a typical time step, an elastic trial state is first computed for prescribed strain in- 
crements and converged internal variables. Then, in the corrector stage the actual stress 
is obtained by the closest-point projection of the trial stress state onto the elastic domain. 
This projection is computed locally at each quadrature point of a typical finite element, and 
depends exclusively on the functional form adopted by the yield criterion in stress space. 
For the J 2 flow theory, the closest-point projection reduces to the classical radial return 
mapping. 

There are two methods to implement the hyperelastoplastic model at finite strains: 
(i) multiplicative decomposition of the deformation gradient F (Simo [1988a] and Simo 
[1988Z?]), and (ii) the spectral form of the right Cauchy-Green tensor C (Ibrahimbegovic 
[1994], Miehe [1998a], Betsch and Stein [1999]). Our numerical experiments show that 
while both methods lead to the same computed results, the second method does not require 
the expensive polar decomposition. For method (ii), we provide a more convenient way to 
decompose the right Cauchy-Green tensor C. 
5.2.2.1. Multiplicative decomposition of the deformation gradient F 

As already discussed in previous section, the present locking-free solid-shell element 
formulations are based on the modified Green-Lagrangian strain E. Therefore, a modified 
deformation gradient F, which is consistent to the modified strain E, is required for the 
algorithm of the finite strain elastoplasticity (e.g., Simo [1988&]). The evaluation of the 
modified deformation gradient F consistent with the modified strain E, however, is rather 



184 



time-consuming. 

The compatible deformation gradient F c from the displacement field is split into an 
orthogonal rotation tensor R and a right-stretch tensor U c as follows 

F C = RU C , (5.14) 

substituting (5.14) into the compatible Green-Lagrangian strain E c of (3.12) 

E c = l - (F cT F c - l) , (5.15) 

the orthogonal rotation tensor R drops out and the compatible stain E c will depend solely 
on the right-stretch tensor U c as 

E c = I [U CT (R T R) U c - l] = l - (U CT U C - l) , (5.16) 

where U c can be obtained by using a polar decomposition on (5.16), and then used in 
(5.14) for the rotation tensor R by the following 

R = F C [UT 1 ■ (5.17) 

The right-stretch tensor U, consistent with the current strain E including the EAS 
and ANS treatments, can be computed by employing another polar decomposition on the 
following 



E = \ (U T U - l) 



(5.18) 



With the unmodified orthogonal rotation tensor R calculated from (5.17) and the 
current right-stretch tensor U obtained from (5.18), the deformation gradient F consistent 
with the current strain E is computed through the following 

F = RU. (5.19) 

Thus, the above procedure to evaluate the deformation gradient F, which is consis- 
tent to the current strains E, needs twice a time-consuming polar decomposition at each 



185 



integration point. We refer to Simo and Hughes [1998, p.244] for the closed-form of the 
polar decomposition. 

With the deformation gradient F, it is straightforward to implement the return map- 
ping algorithm for the J 2 finite strain plasticity model. For more details, readers refer to 
Simo and Armero [1992]. 

5.2.3. 1 . Spectral form based on the right Cauchy-Green tensor C 

Recently, the more advantageous algorithm of isotropic hyperelastoplasticity is based 
on the principal stretch (Simo [1992], Miehe [1998c]), in which all the algorithms previ- 
ously developed for the small-strain plasticity can be used directly. 

The phenomenological description of large-strain elasto-plasticity relies on the local 
decomposition of the deformation gradient F, 

F = F e F p , (5.20) 

where F e is the deformation caused by the stretching and rotation, F p the deformation 
associated with the plastic flow (see e.g., Maugin [1992, p. 167]). 

To construct the nonsymmetric tensor Z e associated with elastic stretches, we de- 
compose the right Cauchy-Green tensor C similar as in (5.20), namely 

C m Z e C p with Z e := C (C p ) _1 , Z eT := {C p )~ 1 C , (5.21) 

where C p := F pT F p is the plastic Cauchy-Green tensor. 

To represent the constitutive relation in terms of the eigenvectors JVj associated with 
Z e , the standard eigenvalue problem needs to be solved 

(Z e - A 2 l) Ni = , (5.22) 

where eigenvalues {A, 2 }- =1 23 represent the elastic principal stretches, and eigenvector Ni 
and covector AT J satisfies 

Ni-N s = S{, ij = 1,2,3, (5.23) 



186 

and N* is normalized with respect to the plastic Cauchy-Green tensor C p 

N^C^N 1 = 1 , i = l,2,3, (5.24) 
which leads to simple expressions of C p and C v ~ l as follows 

C p = j2N i (S>N i , (5.25) 

i=l 
3 

C v ~ l = ]T N* $ JV* , (5.26) 

i=l 

respectively, where the summation convention is not used in the section. 
From (5.22), we can express Z e in the following form 

3 

Z e = '£\?N i ®N i , (5.27) 

i=l 

therefore, with (5.25) and (5.27), the spectral form of C becomes 

3 

C = Z e C p = £ XfNi <g> JV< . (5.28) 

i=l 

Assuming the isotropic free energy if) takes the form 

V> = V(ei,e2,e 3 ,/i) , (5.29) 

where the logarithmic elastic principal strain e t is defined as the function of elastic principal 
stretches Aj, namely 

e< := InA, = hnXf , i = 1, 2, 3 , (5.30) 

and is the equivalent plastic strain. 

Define the corresponding principal stress r< as follows 

r ' := ^' »- 1.2,3. (5.31) 

According to (5.27) and (5.28), the derivatives of A? with respect to Z e and C are 
respectively 

<9A 2 <9A 2 

^^rtQNi, -^L = N i ®N i . (5.32) 



187 



With (5.30), (5.31) and (5.32) 2 , the stresses S in the spectral representation are 

= dip 
dE dc 

m 2\^—— = 2Vr— — = — iV* ® AT (5 33) 

The Kuhn-Tucker loading/unloading condition is 

7 >0, 0<O, 70(r : ,r 2 ,r 3 ,?/) = O, (5.34) 

where is the yield function, 7 plastic parameter, and y = ~. The flow rule for the 

oh 

internal variables h and C p in spectral form can be expressed as the ordinary differential 
equations with respect to the time 

d<j) 

h = Tjj, (5.35) 

C p ~ l = -2 1 C*- l "£^N i ®N i . (5.36) 

i=l 

Remark 5.2. As mentioned in Simo [1988a], the Kuhn-Tucker optimality conditions 
and constraints on the evolution equations for the internal variables can only be obtained 
naturally through the principal of maximum plastic dissipation. 

Consider the following local dissipation inequality (Simo and Hughes [1998, p.229]) 



P = \ S -- b -Jt^ C ^^ h ) 



2{ b 2 dC ■ dC^ ~dh k -°- (537) 



Since 



S=^L = 2— 
dE dC ' 



(5.38) 



and denote y := — , (5.37) is rewritten as 
oh 



V-=-g^--C P ~ 1 -yh>0, (5.39) 



188 



taking the derivatives of eigenvalue problem (5.22) while keeping C fixed, along with 
(5.2 1) 2 , it becomes 

(< 



' CdC p ~ x - ^h dCP ~ 1 ) N * + ( CCP ~ l ~ \ 21 ) dNi = 0, (5.40) 

by multiplying JVj at both sides of (5.40), using the orthogonality condition JVj • dNi = 0, 
and with (5.28), we have 



dC p ~ 1 

it follows that 



XtNiQNi, (5.41) 



dip 



1 - t&Mx-t&x***- w 



dC v ~ l £j d\t dC*- 1 fr[dX\ 
With (5.32) 2 , we have 

dib A dip dXf JL, 0d> , 

— = V— — - = V — AT 1 g> A/" 1 (5 43) 

Therefore, with the above (5.42) and (5.43), and combining the expression of C in (5.28), 
C p in (5.25), and S in (5.33), we obtain the relation 

= C^C = l -CSC p . (5.44) 



dC p ~ l dC 2 
With (5.33) and (5.28), CS in (5.44) becomes 



3 

cs = Yl T i N i® Nl . ( 5 - 45 ) 

i=l 



thus the dissipation inequality in (5.39) can be rewritten as follows 

1 3 

P = --J^TiNi ® APC P : Cf" 1 - yA > . 



2 i= i 



(5.46) 



The classical rate-independent plasticity model in the spectral form is obtained by 
postulating the following local principle of maximum dissipation: 

For admissible (Ti, h), which satisfies the yield criterion (f> (r^, h), the actual state 
(Ti, h) maximizes the dissipation function V. Using the Lagrange multiplier method, we 
define the constrained minimization problem in the following 

C (r,, y, 7) := -V (r<, y) + 7 (r<, y) , (5.47) 



189 



where 7 is the plastic consistency parameter. By finding the extremal point of C, the con- 

dC 

strained minimization problem of (5.47) leads to the flow rule in (5.35) (i.e., — = ) and 

dy 

(5.36) (i.e., — = ), and the Kuhn-Tucker loading/unloading condition of (5.34). I 

OTi 

For the evolution equation, the plastic hardening variable h is obtained by a standard 
implicit backward Euler method with the time increment At = t n+ \ — t n on (5.35) 

d({> 



K+i = h n + P , 
dy 



, = ln+1 At . (5.48) 

n+l 



As originally proposed by Weber and Anand [1990], we integrate the flow rule of 
C p_1 in (5.36) by means of the exponential map. The advantage of this method is two 
folds: i) preserve the plastic incompressibility condition of the pressure insensitive yield 
function 0; ii) return mapping algorithm of small strain plasticity can be used without any 
modification. 

By using the backward Euler integration with an exponential map on (5.36) (simi- 
larly, for y = c(t)y, it follows that y n+ i = [exp (c n+ i At)] y n ), it yields 

C p n ~ + \ = exp (-20 E f^AT ® n] C*T 1 , (5.49) 

V i=l OT i /„+! 

where C v ~ l is associated with the eigenvalue problem (5.22) for the trial state, that is 

(C n+1 CT 1 - A?l) N\ = , (5.50) 

which gives 

C n+l Cl~ x = £A?iVj S N u . (5.51) 

i=l 

In parallel, C„+\ is associated with the eigenvalue problem at t n+ i 

(C n+1 C p n -\ - Xfl) N t = , (5.52) 

which gives 

Cn+iCHl^^Ni^N*, (5.53) 



i=l 



190 



where iVj is the eigenvectors associated with the eigenvalue A* for time £ n+1 . 

Similar to (5.25) and (5.26), the plastic Cauchy-Green tensor C p n and its inverse at 
the trial state are expressed as follows 

3 3 

C P n = E N \ ® N \ . CT 1 = E ^ ® ^ • (5-54) 

i=l i=l 

Substituting (5.54) 2 into (5.51), and recalling (5.28), we have 

3 3 

C n+1 = E AfiV* ® N* = E A i ^ « ^ • (5-55) 



Therefore, the eigenvectors N t at t n+x and jV{ at the trial state are proportional 

\t 

Ni = ^-N\ , (5.56) 

by multiplying AP and JV" at both sides of (5.56), and with (5.23), the eigenvectors N l 
associated with A t at t n+l and AT* 1 associated with X\ at the trial state are also proportional 

N l = ^ t N u . (5.57) 
Considering the above (5.57) in (5.54) 2 , we have the following 

3 3 \t2 

CT 1 =E^® = E T2 N ' ® N * . (5-58) 

i=l i=l A z 

substituting (5.58) into (5.49) and using (5.26), it becomes 

\ 2 - ex P (-2/?f^) A? , (5.59) 

with the definition of (5.30), the update of the principal strain e* is by simply taking the 
logarithm on both side of (5.59), namely 

« (5-60) 



Denote 



{e t }, ^-^},-r-{ ? }-|^| l i = 1,2,3, 



191 

d(j> d(j> dtp 

or Oy Oh 

_ d 2 j) _ dr_ ^ _ d 2 ip _ dy_ 
dede de ' h dhdh dh ' 

d 2 (f> dn d 2 (j) dn h 

= a a = "H - ' -^h = a r, — i p.Ol) 
or or Or oyoy oy 

the solution of the set of nonlinear algebraic equations (5.48), (5.60) along with the yield 
criterion cf> = is typically obtained by a Newton procedure. A Newton procedure based 
on the systematic linearization of these equations gives rise of a plastic-corrector return 
to the yield surface based on the concept of closest-point projection. We write the plastic 
updates in (5.48), (5.60) and yield condition (ft = at time in the form of 

r t = e - e l + 0n = , (5.62) 

r h = h - h l + Pn h = , (5.63) 

(f>(r,y) = 0, (5.64) 

linearizing (5.62)-(5.64) gives the incremental form at iteration (k) of time t n+ y, 

r e + Ae + A(3n + pAn = , (5.65) 

r h + Ah + Apn h + pAn h = , (5.66) 

4> + nAr + n h Ay = . (5.67) 

By making use of the following from (5.61) 

At = £Ae , Ay = S h Ah , (5.68) 
An = TAr , An h = F h Ay , (5.69) 
and substituting (5.68) into (5.67), substituting (5.69) into (5.65) and (5.66), it yields 

r e + £~ l SAe + Af3n = , (5.70) 



192 



r h + S^£ h Ah + Af3n h = , (5.71) 

<t> + n£Ae + n h £ h Ah = , (5.72) 

with I = [E-'+p^y 1 , 4 = fa 1 +pJ r h)~ 1 , (5.73) 
then we can solve for the increments A/3, Ae, Ah from (5.70)-(5.72) as follows 

A/3 = (</> - n T S r e - n h S h r h ) , (5.74) 

Ae = -5" 1 5(r f + A(3n) , (5.75) 

Ah = S^S h ( r h + A(3n h ) , (5.76) 

with D = n T £n + E h n 2 h . (5.77) 

With the above obtained increments, we update the strains, internal variables and plasticity 
parameter for the iteration [k + 1) at time t n+ i. The local Newton iteration procedure is 
continued until the convergence to the yield surface within a sufficient tolerance. 

The Newton's method relies on the fourth-order consistent tangent moduli C (also 
called the algorithmic tangent moduli) , which is crucial in the development of the material 
tangent stiffness matrix (see Section 3.3). In Simo and Taylor [1985], it was first shown 
that the disappointing rates of convergence exhibited by Newton-type iterative methods 
arise from lack of consistency between the continuum elastoplastic moduli and the return 
mapping algorithm. The development of the consistent tangent moduli is based on a sys- 
tematic linearization of the stress update algorithm, which is defined as 

where S at t n+ i is obtained from (5.33). 

Since the principal stress T t at time t n+ i is the function of t\, with (5.32) 2 and (5.33), 
the derivation of (5.78) leads to the algorithmic tangent moduli C (see also Ogden [1984], 
Zienkiewicz and Taylor [2000b, p. 343] for the details) in the following 

^ . Ti N li ® N ti 



c = Y^2 d ^ i J^ ) N ti ®N ti + Y. 2 



t-i dC ' £"A? dC 



193 




N u <g> N u ® N tj ® AT' J 



i=l j=lj#i 

where the elastoplastic moduli is defined in the eigenspace as follows 

which can be derived based on the linearization of (5.60), (5.48) along with = 0. The 
elastoplastic moduli £*? is derived in Appendix A.5, and the term g tJ listed in (A.48). 

Since we develop the present solid shell formulation in the convected basis, it is nec- 
essary to transform the second Piola-Kirchhoff stress tensor S and the consistent tangent 
moduli tensor C from the basis of eigenvector N u to the basis G u namely 

S = S%N tl ®N t] = S ij Gi®G 3 

C = C? jkl N u ®N tj ®N tk ®N tl =C ljkl G i ®G j ®G k ®G l , (5.81) 

which involve the transformation procedure similar to that used between the Cartesian basis 
e l and the convected basis G{ in Section 3.3. 

In summary, we identify {C, C p , h} as the state variables. Once the state variables 
are known, the stress tensor S and the consistent tangent moduli C will be determined 
through (5.33) and (5.79) respectively. The detailed implementation on the stress update 
and consistent tangent moduli are listed in Appendix A.4 and Appendix A.6, respectively. 
5.3. Explicit Time Integration Method for Solid-Shell Elements 

Here we discuss the conditional-stable explicit method associated with the present 
solid-shell element, in which the reliable full numerical integration is employed. Since 
the maximum stable time-step size At decreases with the mesh refinement (see, e.g., Be- 
lytschko, Liu and Moran [2000, p.314], LS-DYNA [1998, Chap. 19]), it is beneficial to use 
the present element, which has coarse mesh accuracy, and needs only one element through 
the thickness, in explicit analyses of shell structures. Reduced-integration (RI) elements 



194 

have been widely used together with explicit time integration (DYNA3D [1993], ABAQUS 
[200 1 ]). Due to the use of ad hoc assumptions on the kinematics and the material properties, 
the proper stabilization (hourglass control) techniques for the spurious zero-energy modes 
caused by the RI scheme are, however, still an active research area, especially for physi- 
cally nonlinear problems such as crashworthiness problems (Zhu and Cescotto [1 996], Zhu 
and Zacharia [1996], Zeng and Combescure [1998]). Furthermore, reduced-integration 
elements are highly sensitive to mesh distortion (see, e.g., Stanley [1985]). 

Although there is a disadvantage with the need for using small time increments for 
stability, the advantages of explicit methods are significant in that the construction, stor- 
age, decomposition and back substitution of the effective tangent stiffness matrix, which 
is required in implicit methods, are completely avoided. There are many situations where 
explicit methods are preferable. For example, in high-speed events such as car crash simu- 
lations, a small time step is required due to the noise introduced by the contact and impact 
between different structural parts. 

Two types of mass matrices can be considered: The non-diagonal consistent mass 
matrix and the diagonal lumped mass matrix. To increase computational efficiency, lumped 
mass matrix is often chosen to avoid the decomposition and back-substitution. Although 
the procedures for diagonalizing the mass matrix are quite ad hoc and questionable, espe- 
cially for high-order elements and stress-resultant shell elements (Zienkiewicz and Taylor 
[1989, p.605], Hughes [1987, p.565]), the lumped element mass matrix for the present 
eight-node solid-shell element can be obtained easily without problem. One common ap- 
proach is the row-sum technique (Zienkiewicz and Taylor [1989, p.474]), in which the 
diagonal entries of the lumped element mass matrix raU are obtained by 

ndof 

m \i = E K . (5.82) 

j=i 



where the sum is over the entire row of the consistent element mass matrix [m c 1 e 
^ndofxndcrf where ndof is the number of degrees of freedom in an element (see Vu-Quoc 
and Tan [20026]). 



195 



Alternatively, the lumped element mass matrix 
from the following expression 



rrii 



= J NpdV , 

n(«> 



can also be evaluated directly 



(5.83) 



where in the element domain Bq \ N is the matrix containing the shape functions corre- 
sponding to degrees of freedom of the element nodes, and p the mass density. Both (5.82) 
and (5.83) lead to the same lumped element mass matrix for the present element. 

The global semi-discretized nonlinear ordinary differential equation (ODE) is in the 

form 



Mu = F ext - F int , 



(5.84) 



where M is the assembled mass matrix, and the internal force F is computed as follows 



-tint 



nel / , . 
~ \ J stiff K ua ^aa J 



stiff ^ua \ r "aa\ J EAS ) ' 



t(e) 



(5.85) 



with the matrices fc^, k^ a , fg\ s being the terms associated with the EAS method in 
(3.78), and EAS parameters are updated through (3.79). 

To solve the nonlinear ODE (5.84), the effective and most widely used explicit 
method is the central difference method. In a typical time step [£ n ,£ n+ i] with time-step 
size t n+ i — t n = A£, the displacement u n+i at t n+ i is updated by using the displacement 
u n , the velocity u n , and the acceleration u n at t n as follows 



At 2 

u n+1 = u n + u n At + u n —— 



By using (5.84) and (5.86), the acceleration u n+1 at time t n+ i is obtained via 



u n+l = M- 1 F ext (vi) - F™ (u n+1 ) 



(5.86) 



(5.87) 



where only divisions are required for the solution of u n+ i due to the diagonal form of 
M, in contrast to the expensive decomposition and back-substitution needed in implicit 



196 

methods such as the Newmark and energy-momentum conserving algorithms (Vu-Quoc 
and Tan [20026]). 

With u n+l from (5.87), the velocity u n+1 at t n +i is approximated by 

w n+ i = u n + -y (u n + u n+ i) . (5.88) 

Updating for the displacement u n+l by (5.86) does not require the solution of any 
algebraic equations. Thus, in this sense, explicit integration is simpler than implicit inte- 
gration. As shown in Appendix A.3, the explicit program is a straightforward evaluation of 
the governing equations and the time integration formulas. It can be seen in (5.85) that the 
computation of the internal nodal forces involves the calculation of strains, stresses, and 
the constitutive matrix. When the element nodal forces are calculated, they are assembled 
to the global array according to the node connectivity. By prescribing the nodal velocities 
at prescribed velocity boundaries (see, e.g., in Appendix A.3), the correct nodal displace- 
ments result from (5.86). The reaction forces at prescribed velocity nodes can be obtained 
from the total nodal forces 

Rn +l = F& - FJJ, • (5.89) 

Since the time step in explicit integration must be below a critical value (otherwise 
the numerical solution will blow up), it is not recommended to use explicit methods in 
quasi-static or low-speed events such as the springback effect in metal forming process. 
On the other hand, it is appropriate to combine explicit methods and implicit methods to 
maximize computational efficiency, that is, for example, in metal forming or crash analysis, 
explicit time integration can be used for the initial time stepping and then a static or implicit 
dynamic solution be used for the rest (see, e.g., ADINA [2002]). 

5.4. Numerical Examples 

The finite element formulation of the present low-order solid-shell element for non- 
linear analysis of shell structure presented in the previous sections has been implemented in 
both the Finite Element Analysis Program (FEAP), developed by R.L. Taylor (citef :zie.89a), 



197 



and the NIKE3D by the Lawrence Livermore National Laboratory (NIKE3D [1995]), and 
run on a Compaq Alpha workstation with UNIX OSF1 V5.0 910 operating system. In each 
element, the mass matrix is evaluated using the Gauss integration 2 x 2 x 2, the tangent 
stiffness matrix, the dynamic residual force vector are evaluated using the Gauss integration 
2 x 2 in the in-plane direction, three Gauss points in the thickness direction for Mooney- 
Rivlin material, and five Gauss points in the thickness direction for elastoplastic material. 
Below we present the examples involving the static and dynamic large deformation analysis 
with nonlinear materials. In the nonlinear dynamics analysis, we use the trapezoidal rule 
for implicit time integration (with the consistent mass matrix) and the central difference 
method for explicit time integration (with the lumped mass matrix), without introducing 
numerical dissipation. 
5.4. 1 . Large Deformation of Rubber Shells 

Here we analyze the large deformation of rubber shells by using the present solid- 
shell element and Mooney-Rivlin material model. For the augmented Lagrangian method, 
a function of 7lnJ 3 is appended to the strain energy function W s in either (5.1) or (5.11), 
where 7 is augmented Lagrangian multiplier. The augmented Lagrangian procedure can be 
accomplished by a nested iteration algorithm (Zienkiewicz and Taylor [2000a, (p.323)]). 
Within a typical iteration of one step, we solve the linear system with fixed penalty pa- 
rameter A. Then the Lagrangian multiplier 7 is updated by 7 = 7 + Am/ 3 at the end of 
nested iteration . The nested iteration can be repeated with the increased penalty parameter 
to reach the desired accuracy for the incompressibility constraint. Here we use an initial 
value of A = 250 for the penalty parameter. Our numerical experience on the uniform 
extension/compression test of one element (Bathe [1996, p.593]) showed that with four 
nested iterations the incompressibility can be enforced to the tolerance of 10 -8 in the vol- 
ume change between the initial configuration and the deformed configuration, compared to 
10~ 5 by using the penalty method with a large value of penalty parameter A = 5000. 



198 

5.4. 1.1. Stretch of a rubber sheet with a hole 

This problem has been analyzed by Gruttmann and Taylor [1992]. The material con- 
stants for Mooney-Rivlin model are C\ = 25 and C 2 = 7. The length of the square is 
L = 20, the radius of the circle is R = 3, and the thickness h = 1 (left of Figure 5.2). Due 
to the symmetry, only one quarter of the sheet has been modeled with 64 solid-shell ele- 
ments. The augmented Lagrangian method is used here, with five nested iterations for each 
load step. Figure 5.2 depicts the initial geometry and the corresponding final deformed 
mesh configuration for q = q = 90. A full agreement with the membrane element devel- 
oped by Gruttmann and Taylor [1992] is shown in left of Figure 5.3. It is clear that the large 
strains and the thickness stretching are involved in the present problem. For instance, the 
sheet thickness at point D becomes one half of the initial thickness. It is interesting to note 
that the sheet thickness at point B is increased rather than decreased (right of Figure 5.3). 




Figure 5.2. Stretch of a rubber sheet with a hole: Initial configuration (left) and deformed 
shape at q Q = 90 (right). 

5.4. 1.2. The snap-through of a conic shell 

Next we show the robustness of present solid-shell elements with large elastic strains. 
This problem was appeared in Li, Hao and Liu [2000] for the application of their meshless 
method, in which 12,300 particles with three particles in the thickness are used. A total 
of 1,800 solid-shell elements (3,720 nodes) are used in the discretization, with only one 
element in the thickness direction. It is noted that the ongoing intensive research is directed 



199 




Figure 5.3. Stretch of a rubber sheet with a hole: Load-displacement diagram (left) and 
thickness stretching (right). 

to make meshless methods more computationally efficient, which includes the interpolation 
scheme, numerical integration procedures and techniques of imposing boundary conditions 
(De and Bathe [20016], De and Bathe [2001a], Atluri and Shen [2002]). 
The material and geometric properties are 

Ci = 18.35, C 2 = 1.468, 

A = 1.468 x 10 3 , p = 1.4089 x 10" 4 , 

Rtop = 1, Rbot = 2, H - 1, h = 0.05 , (5.90) 

where C\, C 2 are the material constants, A the penalty for incompressibility, p the initial 
density of the rubber shell, and R top , R bot the radius at the top and the bottom of conic shell, 
respectively, H the height, and h the thickness of the conic shell. 

The time increment is chosen as At = 25 x 10 _6 sec, and total 400 steps are used 
in the implicit Newmark method. In the computation, we fix the bottom edge of the conic 
shell and do not allow the horizontal movement of the upper inner edge, and prescribe 
the vertical displacement of upper inner edge such that it drags the whole shell structure 
down. At the end of the computation, the conic shell turns inside out. In Figure 5.4-5.5, 
several snap-shots are taken to form a deformation sequence. The deformation is large and 
involves both material nonlinearity and geometry nonlinearity. Such deformation process 



200 



belongs to a so-called snap-through instability problem because the reaction force at top 
edge is alternating along with the advance of vertical displacement of the top edge (see 
Figure 5.6). In the static case, the magnitude of the reaction force is much smaller than in 
the dynamic case, which is true when the inertial effect is not considered (left of Figure 5.6). 



'^7777777777:' 

Figure 5.4. Snap-through of the conic shell: Initial configuration (left) and deformed shape 
at t = 2.5 x 10~ 3 sec (right). 




Figure 5.5. Snap-through of the conic shell: Deformed shape at time t = 5.0 x 10 3 sec 
(left) and t = 10.0 x 10~ 3 sec (right). 

Finally, we point out that, both the penalty method and augmented Lagrangian method 
leads to similar behavior (right of Figure 5.6) in this problem, while the latter needs more 
iterations at each time step (Table 5.1). 
5.4. 1.3. Large motion of the pinched cylindrical shell 

This problem was also appeared in Li et al. [2000] for the application of their mesh- 
less method. We prescribe the inward radial displacement for two opposite nodes of the 
inner-surface at the middle section of the cylinder (Figure 5.7). 



Disp. control 




201 







k 


sialic | | 

u 


Li 1 iji 1 




n 






Figure 5.6. Snap-through of the conic shell: Deflection versus reaction force curve by 
penalty method (left) and augmented Lagrangian method (right). 

Table 5.1. Snap-through of the conic shell: Convergence results for both penalty method 
and augmented Lagrangian method at a typical time step (energy norm) 



Iter. 


Penalty 


1 st augm. 


2nd augm. 


3rd augm. 


4th augm. 


5th augm. 





3.20E-02 


1.44£-03 


1.32£-05 


2.64£-06 


4.74£-07 


8.30£-08 


1 


7.19£-03 


7.35£-03 


2.34E-08 


1.20£-10 


8.30£-13 


8.18E-15 


2 


4.86£-05 


2.80£-06 


2.48£-16 


1.97E-20 


2.86£-24 


5.41.B-28 


3 


1. 16£-08 


2.10J5-12 










4 


9.02E-16 


3.02£-23 











The material properties are the same as in (5.90) and geometric properties are 

R = l, H=l, h = 0.02, (5.91) 

where R is the radius, H the height, and h the thickness of the cylinder. 

Because of the symmetry, only one-eighth of the cylinder needs to be modeled by 
32 x 32 x 1 solid-shell elements with 2,178 nodes (Figure 5.7). For the temporal integration 
of explicit central difference algorithm, the time increment is At — 0.5 x 10 _6 sec and total 
20,000 time step has been taken to finish the run. For the integration by implicit Newmark 
algorithm, the time increment is At = 10 x 10 -6 sec and total 1,000 time step has been 
taken. In Li et al. [2000], total 30,300 particles with three particles in the thickness are used 
and 21,000 time steps are taken in the explicit analysis. 

Both the implicit method and the explicit method lead to the similar deformation 
sequence shown in Figure 5.8- 5.9, The deformation of cylinder under pinched loading 
is drastic. At the end of our computation, the two opposite points of inner surface of the 



202 



cylindrical shell come together. Compared to Li et al. [2000], the deformation from the 
current calculation is more severe and appears more buckling modes. 

, FE model 




H 



Figure 5.7. Large motion of pinched cylindrical shell: Geometry and loading. 




Figure 5.8. Pinched cylinder: The deformation at time t = 2.0 x 10 3 sec (left) and 
£ = 4.0 x 10~ 3 sec (right). 

The relation between the reaction force and the deflection both at the pinch point 
is presented in Figure 5.10. It is observed that for both the implicit method and the ex- 
plicit method, the results of reaction forces are very close, which justify the use of explicit 
method in the present solid-shell element for the high-speed dynamics. 



Figure 5.9. Pinched cylinder: The deformation at time t = 8.0 x 10 3 sec (left) and 
t = 10.0 x 10 _3 sec (right). 

5.4.1.4. Rubber hemispherical shell 

The clamped rubber hemispherical shell is subjected to a point load at the pole. This 
problem is used to demonstrate the localized effects of thickness change and importance 
of applying the external surface loading for moderately thick shell problem. Without any 
modification on the element formulation, the surface loading can be considered naturally 
in the present solid-shell element. 

The material and geometric properties are 

C, = 25, C 2 = 7 , 
A = 1. x 10 3 , 

R = 26.3, h = 4.4 , (5.92) 

where C\, C 2 are the material constants, A the penalty for incompressibility, respectively, 
and R the radius, and h the thickness of the spherical shell. Due to the symmetry, one 
quadrant is modeled by 192 solid-shell elements (Figure 5.11). Displacement control is 
used to drive the top of the hemisphere down. Two different loading cases, top surface 
loading at point A and both surfaces loading at point A and B to approximate the midsurface 
loading in classic shell element (with inextensible director), are considered. 

The both-surface loading keeps the thickness unchanged between point A and B, and 



204 



0.035 1 1 1 1 1 1 r 




-0 005 1 1 1 1 1 1 1 1 1 1 1 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 

It 

Figure 5.10. Large motion of the pinched cylindrical shell: Deflection versus reaction force 
curve. 

produces a "kink" at point B, which is true only for thin shells (left of Figure 5.12). The top- 
surface loading exhibits the anticipated localized effects and the thickness change by about 
70% between point A and B, which produces the physically more reasonable deformation 
(right of Figure 5.12). 

The relation between the deflection at loading location and the corresponding total 
force to produce such deflection for both cases is presented in Figure 5.13. The difference 
on two loading cases shows the importance of localized thickness change effects. 
5.4.2. Large Deformation of Elastoplastic Shells 

For finite deformation J 2 plasticity, the free energy function is assumed to take the 
uncoupled form as follows 

1> (ei, e 2) es,h) = ^A p j + fij^ (e,) 2 + hih 2 , (5.93) 

where A and \i are the Lame parameters, H is a parameter which characterizes the isotropic 
hardening in the material, and h the equivalent plastic strain. The Von-Mises yield criterion 



205 




Clamped 



Figure 5.1 1. Rubber hemispherical shell: Mesh, boundary conditions and loading for the 
one quarter of shell. 





Figure 5.12. Rubber hemispherical shell: Displacement loading at both surface shows 
"kink" at point B (left), and displacement loading at top surface has the smoothness at 
point B. The thickness between point A and B is changed by approximately 70% (right). 



is expressed in terms of principal stresses as 



<f>(T U T 2 ,T 3 ,y) = 



1 



EW- [En 



t=l " \i=l 

where r y is the initial flow stress, y = Hh. 



{t v + v) 



(5.94) 



206 



450 



— top surface loading 

— both surface loading 



400 












2 



4 



6 
W 



8 



10 



12 



Figure 5.13. Rubber hemispherical shell: Load-deflection curve showing the difference of 
the two loading cases. 

5.4.2. 1 . Bending of a cantilever beam 

This example investigates an elastoplastic cantilever beam illustrated in Figure 5.14 
for several aspect ratios L/h, and confirms the correctness of the current solid-shell element 
for elastoplastic material. The mesh for the plate is 20 x 1 x 1. 

The material properties are 



where E, [i, r y , H are the Young's modulus, the Poisson's ratio, the initial yield stress, and 
the isotropic hardening parameter, respectively. 

Numerical results are compared to Eberlein and Wriggers [1999] where the solutions 
are computed from the classical 5-parameter shell element (volumetric locking does not 
appear in elements with the plane-stress condition (Hughes [1987, p. 193])). The good 
agreement in all range of aspect ratios can be observed. For the aspect ratio of L/h = 100, 
the final deformed configuration with the contour of the equivalent plastic strains at the 



£=1.2xl0 7 , /x = 0.3, 



T y = 2.4 x 10 4 , H = 1.2 x 10 5 , 



(5.95) 



207 



plate surface is shown in right of Figure 5.14. 



clamped 





plastic stra 






0.0047 






0,0039 






0026 






0.0013 






0003 




a 


0.0000 



Figure 5.14. Out-of-plane bending: Geometry and mesh of cantilever plate (left), and 
deformed shape and upper-surface contour plot of equivalent plastic strains (right). 

A load deflection curves for different aspect ratio are shown in Figure 5.15-5.16. 
Compared to our seven-parameter EAS formulation, the solid-shell element with five- 
parameter EAS in Miehe [1998&] and Klinkel et al. [1999] behave stiffer and cannot obtain 
the collapse load for the thick and moderate thin plate (Figure 5.15), which may be caused 
by the inability of the element to deal with the volumetric locking in the plastic deforma- 
tion stage. It is observed for the thin case (aspect ratio = 1000) the plastic stage doesn't 
appear till the end of loading, so two Gaussian points through the thickness are enough 
for this elastic problem, and the results from Miehe [1998&] and Klinkel et al. [1999] is 
close to the present though their element formulations cannot pass the out-of-plane bend- 
ing patch test (Subsection 3.5.1). The plasticization appears at load F = 475 for aspect 
ratio L/h = 10, and F = 5 for aspect ratio L/h = 100. 



Cantilever beam loaded by shear lores, L/h • 1 



Cant it eve t beam loaded by shear tores, L/h « 100 



pre sent solid-shell 

O Ebenem etal. [1999] 
— 5-param L AS 



10 




Figure 5.15. Out-of-plane bending: Load deflection curve for aspect ratio L/h — 10 (left) 
and for aspect ratio L/h = 100 (right). 




1 2 3 4 5 6 7 

W 



Figure 5.16. Out-of-plane bending: Load deflection curve for aspect ratio L/h = 1000. 

Since the exact tangent (Jacobian) matrix is employed in the Newton's solution pro- 
cedure, the quadratic rate of asymptotic convergence was actually observed in all problems 
we examined. Table 5.2, which depicts the values of the Euclidean norm of both the resid- 
ual and the energy norm at each iteration, clearly exhibits the quadratic rate of asymptotic 
convergence, which is in the sharp contrast to the extremely slow convergence (hundreds 
of iterations in one step is not uncommon) with the hypoelastoplastic rate-form material 
model (Choudhry and Wertheimer [1997]). 
5A2.2. Elastoplastic response of a channel beam 

A far larger and equally important class of structures are the nonsmooth shell which 
consists of folds, kinks or branches. Here we present the warping of an angle iron to prove 
the validity and applicability of the proposed solid-shell element formulation, in which the 
simulation of such structures on both the overall behavior and local stress concentration ef- 
fect at one time becomes straightforward, and does not introduce any assumption on the in- 



209 



Table 5.2. Out-of-plane bending: Convergence results for plate with L/h = 100 (residual 
norm, energy norm) 



Iter. 


Step 1 (F=0.5) 


Step 10(F=5.0) 


Step 13 (F=6.5) 





2.505-01,8.275-02 


6.835+00, 7.735-02 


1.765+01,7.275-02 


1 


1.005+04, 1.795+00 


9.025+03, 1.455+00 


8.215+03,1.205+00 


2 


3.965+00, 3.295-07 


5.275+00,3.625-04 


4.585+01, 1.53S-02 


3 


3.465-05,3.925-14 


3.83E+01, 2.59£7-05 


2.125+03,8.335-02 


4 


5.515-09,7.105-25 


4.145-03,9.195-10 


8.225+01,1.115-02 


5 




1.305-04,3.015-16 


3.835+03,2.575-01 


6 




1.705-08,7.015-24 


2.515+00,3.415-03 


7 






1.995+03,6.915-02 


8 






1.345-01,5.705-06 


9 






3.405+00,2.025-07 


10 






6.835-06,6.165-14 


11 






4.575-08,3.995-23 



terconnection or penalty parameters for the shell intersection. A cantilever beam subjected 
to a point load is considered as given in Figure 5.17. Because the centroid (inside of the 
open cross section) and the shear center (outside of the open cross section) of the beam do 
not coincide, if the concentrated forces are not applied at the shear center of the cross sec- 
tion, the beam section will twist significantly. Originally, the purely elastic material behav- 
ior of this example was presented by Chroscielewski, Makowski and Stumpf [1992], where 
the numerical results show a strong dependence on a penalty multiplier which accounts for 
the drilling stiffness within their shell formulation. The same example for elastoplasticity 
was also investigated by Eberlein and Wriggers [1999], where the converged results are 
presented for their 6/7 -parameter shell concept along with a penalty method for the shell 
intersection. Since the present element uses the upper and lower surface to describe the 
deformation, the intersection can be simply modeled by solid elements. For the detailed 
analysis of the intersection, a refined solid element mesh can be used, which can be easily 
connected to other shell parts modeled by solid-shell elements. It is noted that it is not 
possible for stress-resultant shell element to describe the detailed strain and stress distribu- 
tion at the intersection (Chroscielewski et al. [1997]), while the solid-shell element offers 



210 



a more realistic and convenient representation for the physical structure. 
The ideal elastoplastic material properties are used as follows 

E = 10 7 , n m 0.333, T y = 5.0 x 10 3 , H = Q, (5.96) 

where E, fi, r y , H are the Young's modulus, the Poisson's ratio, the initial yield stress, and 
the isotropic hardening parameter, respectively. 

For the computations, a discretization of 1,584 elements is used as shown in Fig- 
ure 5.17. In Figure 5.20, the load-deflection curves for both the elastic material and elasto- 
plastic material obtained from the present element coincide very well with solutions by 
Betsch et al. [1996] and Eberlein and Wriggers [1999], respectively. The deformed con- 
figurations for u = 4 are illustrated in left of Figure 5.18 for the elastic solution and right 
of Figure 5.18 for the elastoplastic solution. For the elastic solution, the buckling of the 
upper flange can be observed in the vicinity of the clamped edge, whereas the free end is 
twisted, since the external load does not act in the shear center. The elastoplastic solution 
also shows a buckling phenomenon in the upper flange, but it is more concentrated than 
in the purely elastic case. Since the plastic strains occur very early during the deformation 
process as shown in Figure 5.20, the equivalent plastic strains for the deformed configura- 
tion u = 0.2 are reported in left of Figure 5.19. As expected from the beam theory, there 
is a maximum of plastic deformation in the lower and upper flange at the clamped edge of 
the steel channel. The equivalent plastic strains for the deformed configuration u = 4.0 are 
shown in right of Figure 5.19. 

It is noted that for the current shell with intersection, there are two ways to model 
the intersection as in Mesh 1 and Mesh 2 of Figure 5.21. From our numerical experience, 
the results from the Mesh 2 is stiffer than Mesh 1, which is more pronounced for the linear 
elastic material (Figure 5.20). The possible reason for it lies on the sensitivity of assumed 
strain method to the mesh distortion through the thickness. For folded shells with multi- 
intersections, Mesh 3 in Figure 5.21 is more advantageous for the modeling, compared to 
the more involved shell elements with drilling couples (Chroscielewski et al. [1997]). 





Figure 5.18. Channel beam: Deformed mesh for elastic material at u — 4.0 (left) and 
deformed shape for elastoplastic material at u — 4.0 (right). 

5.4.2.3. Pinched hemisphere 

The present example investigates an elastoplastic hemisphere, which has been calcu- 
lated previously by Simo and Kennedy [1992] by means of a 5-parameter shell formulation 
in combination with a plasticity model formulated in stress resultants, based on an addi- 



212 





Figure 5.19. Channel beam: Deformed mesh and contour plot of equivalent plastic strains 
at u = 0.2 (left) and at u = 4.0. (right). 




- present elastic 
model *Ncoi net eiem 

- elasioptasix: 

model nl comer etem 



U U 

Figure 5.20. Channel beam: Load deflection curve by model with solid elements at corner 
(left), and model without solid elements at corner (right). 

tive decomposition of the strains into elastic and plastic parts. The radius and thickness of 
hemisphere are R = 10 and h = 0.5, respectively. Figure 5.22 illustrated the mesh and 
the boundary condition for the present calculation, where total 432 solid-shell elements 
are used to model the one quadrant of the hemisphere due to the symmetry. The material 
properties are 



E = 10, n = 0.2, r y = 0.2, H = 9, 



(5.97) 



where E, fx, T y , H are the Young's modulus, the Poisson ratio, the initial yield stress, and 
the isotropic hardening parameter, respectively. 

Figure 5.23 shows the deformed hemisphere including an outer-surface contour plot 
of the equivalent plastic strains at a load level of F = 30 x 10~ 3 . The load-displacement 
curves are depicted in Figure 5.24, where the results reported in Eberlein and Wriggers 
[1999] are included. Our results agree with Eberlein and Wriggers [1999] very well. 



213 



"71 



Mesh 1 



Mesh 2 



Mesh 3 



Figure 5.21. Mesh at the cross section of shells with intersection. 




Figure 5.22. Pinched hemisphere: Geometry and mesh of one quadrant of hemisphere. 

5.4.2.4. Elastoplastic response of a simply supported plate 

We consider the elastoplastic deformation of a simply supported square plate. The 
square plate has a length of L = 0.508 and thickness h = 2.54 x 10" 3 . Due to the symmetry 
of geometry and the boundary conditions, only a quarter of the plate has been discretized 
with 16x16x1 solid-shell elements (Figure 5.25). It is simply supported so that horizontal 
displacements and rotations may occur. Only the transversal displacements are set to zero 



214 




plastic stra 




0.0486 




0.0347 




0.0208 




0.0069 




0.0041 




0.0020 




0.0000 




Figure 5.23. Pinched hemisphere: Deformed shape and outer-surface contour plot of equiv- 
alent plastic strains. 




Figure 5.24. Pinched hemisphere: Load F versus the deflection under load F. 

at the boundaries of the upper surface of plate. 

The ideal elastoplastic material properties are 

E = 6.9 x 10 10 , n = 0.3, T y = 2.48 x 10 8 , H = , (5.98) 



Figure 5.25. Inflation of a plate: Mesh, boundary conditions and loadings for the one 
quadrant of square plate. 

where E, fi, T v , H are the Young's modulus, the Poisson ratio, the initial yield stress, and 
the isotropic hardening parameter, respectively. 

A uniform transverse dead-load of p = 10 4 has been applied at the upper surface of 
plate. Figure 5.27 depicts the load deflection curve where the load factor / in p — fp has 
been plotted as a function of the vertical displacement w of the center point of the square 
plate. The curve of the present formulation is in agreement with the converged results from 
the high-order shell element in Schieck, Smolenski and Stumpf [1999], which is available 
up to / = 27. 

Figure 5.26 displays the deflection of the plate at the load level / = 70, along with 
the upper-surface contour plot of the equivalent plastic strains. The remarkably distorted 
element in the corner zones of the plate is observed. And the localization of the plastic 
strains in the four corner zones of the square plate is appeared. 
5A2.5. Elastoplastic response of a pinched cylinder 

The example is concerned with the elastoplastic deformation of a thin-walled cylin- 
der with large equivalent plastic strain (about 100%). This example has already been inves- 



216 




plastic stra 






0.2567 






0.2007 






0.1345 






0.0701 






0.0141 






0.0005 



Figure 5.26. Inflation of a plate: Deformed shape and upper-surface contour plot of equiv- 
alent plastic strains. 




0.16 



Figure 5.27. Inflation of a plate: Load factor / versus vertical displacement w of center 
point of square plate. 

tigated by Simo and Kennedy [1992], Wriggers, Eberlein and Reese [1996], and Eberlein 
and Wriggers [1999]. The geometry and the boundary conditions of the cylinder are dis- 
played in Figure 5.28, where the mesh of 32 x 32 x 1 solid-shell elements are used for one 



217 



eighth of the cylinder due to the symmetry. The cylinder is loaded with two radial pinched 
displacements in the middle of the structure. At both ends the boundary conditions have 
been prescribed such that the circular shape of the end cross section is preserved and free 
deformation in axial direction is allowed. The von Mises-type elastoplastic material with 
isotropic hardening response is governed by the material parameters 

£ = 3000, M = 0.3, 7, = 24.3, # = 300, (5.99) 
where E, (M, r y , H are the Young's modulus, the Poisson's ratio, the initial yield stress, and 
the isotropic hardening parameter, respectively. 

FE mesh 

32 x 32 x 1 

Rigid diaphragm 

L = 600 
R = 300 
h = 3 




H *T — 




1 1 • 




• 

1 1 • 

1 1 1 
■ 

1 1 ■ 

1 . 




1 • 


1 ' ' 


. 1 • 



/ « 

/.* 

/ » 



w 



w 



Figure 5.28. Pinched cylinder: Geometry and loading. 
Figure 5.29-5.30 display the deformation shapes of half of the cylinder at w = 150, 
= 200, w = 250, and w = 280, along with the the contour plot of the equivalent plastic 
strains on the outer surface of the shell, respectively. The maximum equivalent plastic 
strains happened at the loading areas. Figure 5.31 depicts the load deflection curve of the 
problem which has been achieved in an incremental deformation controlled process. The 
curve is in agreement with the Wriggers et al. [1996] within a wide region. 



218 





"Hill 

Figure 5.29* Pinched cylinder: Deformed shape and outer-surface contour plot of equiva- 
lent plastic strains at w = 150 (left) and at w = 200 (right). 




plastc Btra • 






0.6102 
0.4882 






0.3661 






0.2441 






0.1220 






0.0045 






0.0000 





0.9978 
0.7675 
0.5373 
0.3070 
0.0768 
0.0050 
0.0036 



Figure 5.30. Pinched cylinder: Deformed shape and outer-surface contour plot of equiva- 
lent plastic strains at w = 250 (left) and at W = 280 (right). 

5A2.6. Free-flying multilayer plate with ply drop-offs 

This example establishes the capability and performance of the present solid-shell 
in modeling elastoplastic multilayer plates/shells with ply drop-offs. The same model for 
linear elastic material was demonstrated in Subsection 4.4.4. The geometry, the loading 
configuration and the time history loading amplitude are described in Figure 4.9. The mesh 
of a three-layer plate with ply drop-offs is shown in Figure 4.21 (i.e., L = 0.3m, W = 
0.06m, and {t) h - 0.001m). The plate is divided into three equal parts and two ply drop- 
offs along its length, with each part having a length of 0. 1 . We use sixty solid-shell elements 
to model the three-layer plate with ply drop-offs, with a time-step size of At = 25 x 10~ 6 sec 
and a total simulation time of t = 0.08sec. The effect of gravity force is not considered. 

The material properties are 

E = 206. x 10 9 Pa, u = Q.? p= 7S00.Kg/m 3 , r y = 245 x 10 6 Pa, H - 1. X 10 9 Pa , 

(5.100) 



3500 



3000 



2500 



2000 

I*. 

1500 



1000 



500 








50 100 150 200 250 300 

W 

Figure 5.31. Pinched cylinder: Load deflection curve. 

where E, fi, p, r y , H are the Young's modulus, the Poisson's ratio, the mass density, the 
initial yield stress, and the isotropic hardening parameter, respectively. 

Snap shots of the multilayer plate undergoing large overall motion and large defor- 
mation are taken in every 4 x 10 _3 sec, from the simulation using the implicit Newmark 
algorithm without numerical dissipation, and are displayed in Figure 5.32. Furthermore, 
the variation of linear and angular momenta along the time are shown in Figure 5.34. Both 
the linear momentum and the angular momentum are conserved. On the left of Figure 5.35, 
the kinetic energy, the strain energy (computed via (5.93)), the total energy (kinetic + strain 
energies), and the work of the external forces are plotted as a function of the integration 
time. Due to the physical plastic dissipation, the total energy is much less than the work 
of the external forces, without any numerical damping. Unlike in the elastodynamics, the 
strain energy remains on a very low level, which indicates that the free-flying of plate is 
essentially a rigid-body movement (comparing Figure 5.33 to Figure 5.32). For the elasto- 
plastic dynamic analysis, the Newmark algorithm possesses a stable number of iterations 



■ 



220 



and thus a good rate of convergence at each time step (right of Figure 5.35). The smooth- 
ness of both kinetic energy and strain energy along the time suggests a very low level of 
high-frequency "noises" in the response, compared to the elastic case where the introduc- 
tion of numerical dissipation is crucial to suppress the high-frequency modes. 



1 "I- '' 




Figure 5.32. Free-flying three-layer plate with ply drop-offs using the Newmark algorithm: 
Perspective view. 




Figure 5.33. Free-flying three-layer plate with ply drop-offs: Deformed shapes for linear 
material material (left) and elastoplastic material (right) at t = 16 x 10 _3 sec. 



221 




Figure 5.34. Free-flying three-layer plate with ply drop-off s: Linear momentum and angu- 
lar momentum using the Newmark algorithm. 



Strain 
• KrnahoStrain 
- Ealamal Wor* 



0.01 0.02 0.03 0.04 0.05 O.OB 0.07 0.06 

Tmm(mc) 




Figure 5.35. Free-flying three-layer plate with ply drop-offs: Energy balance (left) and 
Number of iterations (right) using the Newmark algorithm. 

Overall, the Newmark method yields a stable integration for the elastoplastic shells. 
It is interesting to note that a more expensive mid-point rule may conserve the angular 
momentum exactly via a product formula algorithm, while the accuracy of the dynamic 
response remains about the same as in using the Newmark method (Simo [1992, Fig. 14]). 
5.4.2.7. The impact of a boxbeam 

In this example, we simulate a boxbeam being impacted at one end while the other 
end being fixed. The rigid impactor is assumed having an infinite mass with a fixed velocity 
of v = 20m/ s (Figure 5.36). For this high-speed problem with rough response, the explicit 
method is suitable. 



222 



The material properties are 

£ = 21. X 10 9 Pa, n = 0.3, p = 7800. Kg/m 3 , r y = 1.06 x 10 9 Pa, H = 40.9 x 10 6 Pa , 

(5.101) 

where E, p, p, r y , H are the Young's modulus, the Poisson's ratio, the mass density, the 
initial yield stress, and the isotropic hardening parameter, respectively. 

Neglecting frictions between the impactor and the boxbeam, and without considering 
the self-contact on the surface of the boxbeam, we assumed that once the impact occurs, 
the rigid impactor stays with the impacted end of the boxbeam, which moves down with 
the same constant velocity v as the impactor, while its displacements in both X-direction, 
and V-direction are constrained. 

Due to the symmetry, only one quarter of the boxbeam is modeled by 960 solid-shell 
elements (2,080 nodes), as compared to 7,952 particles used in meshless method of Li et al. 
[2000]. With time-step size At = 0.4 x 10" 6 sec, a sequence of deformations are displayed 
in Figure 5.37-5.38, where half of the structure is displayed for a better visualization of 
the buckling modes. The experiment results show that the first few buckling modes should 
appear immediately at the impact location (Zeng and Combescure [1998]). Our results 
give the same prediction on the locations where the buckling modes should appear. The 
final deformed shape and the contour of equivalent plastic strains of outer surface of the 
boxbeam at t = 1.6 x 10~ 3 sec is shown in left of Figure 5.39. It is observed that the large 
equivalent plastic strains take place at the corner of the boxbeam. The relation between the 
reaction forces and the deflection at the collision end is shown in right of Figure 5.39. For 
the comparison, the calculation by using the implicit Newmark method was also carried 
out with time-step size At = 4 x 10" 6 sec, and similar results on deformations confirm 
the validity of explicit method with the present solid-shell element in elastoplastic large 
deformation shell analysis. 



223 



Rigid Impactor 





v = 20m/s 

L •■ 203mm 
a = 50.8mm 
b = 38.1mm 

h = 0.914mm 



Y 



X 



h 



Cross section 



Figure 5.36. The impact of a boxbeam: geometry and loading. 





Figure 5.37. The impact of a boxbeam: The deformation at time t = 0.4 x 10~ 3 sec (left) 
and at time t = 1.0 x 10" 3 sec (right). 

5.4.2.8. Pipe whip 

This example is to show the modeling capability of the present solid-shell element 
in contact problems. The element provides a natural and efficient way for shell contact 
problem since double-side surfaces of shell are available and the transverse normal stress 
is included. To authors' knowledge, the shell contact problem using solid-shell elements 



Figure 5.38. The impact of a boxbeam: The deformation at time t = 1.3 X 10 _3 sec (left) 
and at time t = 1.6 x 10" 3 sec (right). 




piastic stra 






0.4834 






0.3443 






0.2052 






0.0662 






0.0087 
0.0005 




Figure 5.39. The impact of a boxbeam: Deformed shape and outer-surface contour plot of 
equivalent plastic strains at time t = 1.6 X KT 3 sec (left), and deflection versus reaction 
force curve (right). 

has not been discussed in the literature before. 

This transient dynamic analysis simulates the impact of two steel pipes without con- 
sideration of friction. The pipes have the outside diameter of 3.3 1 25, the thickness of 0.432, 
and the length of 50. The target pipe is supported with a fixed boundary at each end. The 
impacting pipe swings freely about a point at one end with an angular velocity of 75 radians 
per second (Figure 5.40). The material properties for both pipes are the same 

E = 3.0 x 10 7 , p = 0.3 ,p = 7.324 x 10~ 4 , r y = 7.0 x 10 4 , H = , (5.102) 

where E, fx, p, Ty , H are the Young's modulus, the Poisson's ratio, the mass density, the 
initial yield stress, and the isotropic hardening parameter, respectively. 



Because of the symmetry, only one half of the geometry was modeled, in which 
total 840 solid-shell elements are used with one element in the thickness direction (1,830 
nodes). Total 300 time steps are used with an equal time-step size At = 5.0 x 10" 5 sec 
in the implicit Newmark time integration without numerical dissipation. Slide surfaces are 
defined in potential contact outer surfaces of both pipes. This prevents nodes on one pipe 
from penetrating element surfaces of the other pipe, and allows the contact areas to evolve 
as the pipes deform. Here the slave surface is defined over the region of refined mesh at the 
outer surface of the impacting pipe, and master surface is defined over the region of refined 
mesh at the outer surface of the impacted pipe. 

Figures 5.41-5.42 shows deformed configurations at various time stages of the calcu- 
lation. The contact begins at a single point, and then evolves to two separate regions. Late 
in the analysis, a gap opens at the initial contact point. At the final state of t = 15 x 1(T 3 sec, 
the upper pipe has obviously rebounded. Figure 5.43 shows the contour of equivalent plas- 
tic strain at outer surface of both pipes at the final state of deformation, in which the cross 
section of the upper pipe is severely collapsed along with a local yielded region around the 
contact area. 

Initial angular vel. 




Figure 5.40. Pipe whip: Geometry and loading. 
Since this problem involves dynamics and contact between deformable bodies with 



226 




Figure 5.41 . Pipe whip: Deformed shape at t = 10 3 sec (left) and t = 5 x 10~ 3 sec (right). 




Figure 5.42. Pipe whip: Deformed shape at t = 10 x 10" 3 sec (left) and t = 15 x 10 _3 sec 
(right). 

nonlinear material, it provides a very good illustration of the applicability of the proposed 
solid-shell element to a wide range of problems. 




Figure 5.43. Pipe whip: Deformed shape and outer-surface contour plot of equivalent 
plastic strains, t = 15 x 10 -3 sec. 



CHAPTER 6 

SOLID SHELL ELEMENT FOR ACTIVE PIEZOELECTRIC SHELL 
STRUCTURES AND ITS APPLICATIONS 

6.1. Introduction 

An active shell structure has distributed sensors and actuators, along with the control 
algorithm to control the response of the host structure. The applications range from the 
vibration and buckling control (Berger, Gabbert, Koppe and Seeger [2000], Balamurugan 
and Narayanan [2001a]), to the shape control (Gabbert, Koppe and Seeger [2001]) and 
the noise suppression (Lim, Gopinathan, Varadan and Varadan [1999]). The active struc- 
tures have great potential in the design of light-weight and high-strength structures that are 
widely used in areas such as aerospace (Thirupathi, Seshu and Naganathan [1997], Loewy 
[1997]) and automotive industries (Chopra [1996]). 

In recent years, considerable effort has been devoted to the modeling and control- 
ling of active piezoelectric shell structure, see, e.g., Chee, Tong and Steven [1998], Sunar 
and Rao [1999], Benjeddou [2000] and references therein. The coupled electromechani- 
cal properties of piezoelectric ceramics and their availability in the thin shell form make 
them increasingly popular for the use as distributed sensors and actuators (Niezrecki, Brei, 
Balakrishnan and Moskalik [2001]). The direct and converse piezoelectric effects govern 
the electromechanical interaction in these materials. The direct piezoelectric effect states 
that a strain applied to the material is converted to an electric charge, while the converse 
piezoelectric effect states that an electric potential applied to the material is converted to a 
strain. 

The finite element modeling based on the classical laminate plate theory (Hwang and 
Park [1993]) and first-order shear deformation theories (e.g., Detwiler, Shen and Venkayya 
[1995], Balamurugan and Narayanan [2001/?]) has certain limitations due to their improper 

228 



229 

modeling of the piezoelectric shell structure (Gopinathan, Varadan and Varadan [2000]). 
Improvements are made by using high-order shear deformation theories (Correia, Gomes, 
Suleman, Soares and Soares [2000]) and layerwise shell element formulations (Tzou and 
Ye [1996], Saravanos [1997]), while some shortcomings still remain in that 1) they do not 
consider the transverse normal stress in the element formulation, which may affect the be- 
havior of multilayer structure (Fox [2000]); 2) for the large deformation analysis, the finite 
rotation update associated with rotational degrees of freedom (dofs) in shell formulations 
is very complex to handle (Vu-Quoc, Deng and Tan [2000]); and 3) it is troublesome, if not 
impossible, to incorporate nonlinear piezoelectric material models associated with large 
input signals (e.g., Ghandi and Hagood [1997], Kamlah and Tsakmakis [1999], Kamlah 
and Bohle [2001]) into shell elements based on the plane-stress assumption. 

On the other hand, due to i) the complex geometry, ii) the material anisotropy, iii) 
the coupling of electric field and mechanical field, and iv) the need to satisfy boundary 
conditions of both electric field and mechanical field, 3-D detailed modeling for piezoelec- 
tric shell structures is used extensively. For example, twenty-node solid element in Koko, 
Smith and Orisamolu [1999] and eight-node solid element with incompatible modes in Ha, 
Keilers and Chang [1992] and Panahandeh, Cui and Kasper [1999] were used to model such 
structures. As mentioned in Chapter 3 and demonstrated here, the eight-node solid element 
with incompatible modes and the twenty-node solid element (with reduced integration) are 
not as accurate as shell elements for thin plate/shell problems. To achieve better accuracy, 
excessive number of solid elements are needed (see, e.g., Figure 2.24). To be computation- 
ally economical, Kim et al. [1997] has proposed the use of 3-D solid elements in modeling 
the piezoelectric devices, shell elements for the host structure, and transition elements to 
connect 3-D solid elements in the piezoelectric region with shell elements used for the 
structure. The modeling is complex to handle because of the mixing of several types of el- 
ements and the need for tuning of the aspect ratio of the transition elements. Moreover, the 
use of 3-D solid elements still lead to unnatural stiffening of piezoelectric devices, and as 
a result, artificially high natural frequencies. Alternatively, an assumed-stress piezoelectric 



230 



solid-shell element is proposed in Sze, Yao and Yi [2000], which had coarse mesh accuracy 
and could give accurate results for linear piezoelectric shell analysis; By construction, how- 
ever, the assumed stress method, obtained from the Hellinger-Reissner variational principle 
(Zienkiewicz and Taylor [2000a, p.284]), encounters the difficulty when incorporating the 
classical strain-driven nonlinear material models (Simo et al. [1989]). 

Therefore, the objective of our investigation is to develop efficient and accurate fi- 
nite elements which have the ability to model the nonlinear coupled mechanical-electrical 
response of multilayer composite shell structure which contains distributed piezoceramics. 
In the present work, a piezoelectric solid-shell element is formulated, and active vibration 
control of multilayer plate/shell with distributed piezoelectric sensors and actuators is re- 
alized by the control algorithm of the linear-quadratic regulator (LQR). The simplicity and 
efficiency of solid-shell elements for general nonlinear shell applications were proved in 
our previous work (Chapters 3-5). By including the electric dofs in the solid-shell ele- 
ment, the piezoelectric solid-shell element is developed within the context of the Fraeijs 
de Veubeke-Hu-Washizu (FHW) variational principle. Furthermore, a composite solid- 
shell element is proposed based on the solid-shell element previously developed, which 
can reduce considerably the number of equations, therefore, the amount of memory and 
the running-time, while at the same time keeping the same accuracy for thin multilayer 
composite shells and having the flexibility for the refinement through the thickness. The 
combination of the present piezoelectric solid-shell element and the solid-shell element en- 
ables the analysis of general curved active shell structures in the nonlinear regime, which 
in practice, becomes increasingly important such as in aerospace, MEMS applications. By 
combining the control algorithm, the sensor outputs predicted by the FE analysis could be 
used to determine the amount of input to the actuators for controlling the response of the 
integrated structures in a closed loop. 

The outline of this chapter is as follows: Section 6.2 reviews the kinematics and 
presents variational formulation of the piezoelectric solid-shell element, and discusses 
the composite solid-shell element. Section 6.3 presents the control design for structures 



231 



with piezoelectric sensors and actuators. Numerical simulations which illustrate the per- 
formance of the proposed formulations, including comparisons with available experiment 
results and solutions obtained from shell elements and solid elements, are given in Sec- 
tion 6.4. 

6.2. The Solid-Shell Formulation 
We have developed the solid-shell formulation for geometric and material nonlinear- 
ities based on the enhancement of the Green-Lagrange strain E (see Chapter 3, Chapter 5), 
which leads to particularly efficient computational effort. We briefly describe below the 
kinematics of a piezoelectric solid shell in curvilinear coordinates, and present the piezo- 
electric solid-shell element formulation based on the three field FHW variational principle 
and the EAS method, and then discuss the composite solid-shell element. Readers are 
referred to Chapter 3 for more details. 

6.2.1. The Kinematics of Piezoelectric Solid-Shell Formulation 

To overcome the known problems associated with the rotational degrees of freedom 
in the shell elements, we describe the shell kinematics by linear combination of a pair 
of material points at the top and bottom surfaces of the shell. Therefore, the assumption 
of the shell element that the normals to the element mid-surface remain straight but not 
necessarily normal during deformation, is still adopted. Thus, the initial (undeformed) 
three-dimensional continuum of the shell geometry (Figure 3.2) 

t = e □ := [-1,1] x [-1,1] x [-1,1] , (6.1) 

where X u and Xi are the position vectors on the upper surface and lower surface of solid- 
shell element in initial configuration, respectively, (£\ £ 2 ) the convective coordinates in the 
in-plane direction, £ 3 the convective coordinate in the thickness direction, and □ represents 
the bi-unit cube. 

Similarly, in the deformed configuration, the current three-dimensional continuum is 



232 



described by 



(6.2) 



where x u and x x are the current position vectors on the upper surface and the lower surface 
of the solid-shell element in the deformed configuration, respectively. 

The initial configuration is related to the deformed configuration (see Figure 3.3) by 
the displacement field u as 



x(0 = x(o + w(0 



(6.3) 



The convected basis vectors in the initial configuration are obtained by partial deriva- 
tives of the position vector X with respected to the convective coordinates £' as 

dx(t) 



1 = 1,2,3, 



(6.4) 



which satisfies 



G { • G j = 8{ , dj — Gi'Gj , i,j = l,2,3, 



(6.5) 



where Gij are the metric coefficients of the initial configuration. To simplify the presen- 
tation, we will omit (£) in vectors like Gi($). The covectors G 1 can be obtained by the 
following 



G l = G lJ Gj, with G 13 = [Gij] € 



i-l ,_ 10)3x3 



(6.6) 



and the convected basis vectors in the deformed configuration are defined in the similar 
way, by using (6.3) and (6.4), 



dx du 
g 1 = -^- i =G i + — , 2 = 1,2,3, 



(6.7) 



similar to (6.5) and (6.6), the covector g l , the metric coefficients g%j, and g lj in the deformed 
configuration are obtained. 



233 



By using (6.4) and (6.7), the deformation gradient F relates the basis vectors G t of 
the initial configuration to the basis vectors g { of the deformed configuration as follows 

F=^r=^®GT, (6.8) 

and with (6.5) and (6.8), the compatible Green-Lagrange strain tensor E c is obtained as 

E c = l - (F T F ~h) = \ (9ij ~ Gij) G 1 & - E\p G j , (6.9) 

where J 2 is a second-order metric tensor, Ef s are the covariant components of the strain 
tensor E c . 

The corresponding second Piola-Kirchhoff stress tensor S is expressed in the same 
convected basis G iy i.e. 

S = S %j G l ®G j . (6.10) 

where 5 lJ are the covariant components of the stress tensor S. 

Similar to (6.1), the electric potential <j> is described as follows 

m = \[(i+e)<i>u{e,e)+{i-e)^Ke)] , 

where </> u and fa are the electric potentials on the upper surface and lower surface of solid- 
shell element, respectively. 

The electric field £ is derived from the gradient of the electric potential (p with respect 
to the position vector X as 

5(0) = -GRAD(0) = -^G l . (6.12) 

The corresponding electric displacement vector D is expressed in the same convected 
basis Gi 



D = D l Gi . (6.13) 



234 

6.2.2. Piezoelectric Solid-Shell Element 

Here we present the piezoelectric solid-shell element formulation used for piezoelec- 
tric sensors and actuators in active shell structures. The variational formulation of the EAS 
method, and the finite element approximation of the developed solid-shell elements for the 
multilayer composite shells are described in Chapter 3. 
6.2.2. 1. Functional and finite element formulation 

The electric flux conservation for the piezoelectrics is described as follows 

Yl E = _ J D-SdV - J q v 4>dV - J qsHS , (6-14) 

Bo Bo S q 

where D is the electric displacement vector, S the electric field defined in (6.12), 4> the 
electric potential, q s the electric flux per unit area on the surface, and q v the electric flux 
per unit volume. 

The variation of functional (6.14) with respect to the electric potential is 

SU E = - J D'SEdV- J qvHdV- J q s 5(pdS, (6.15) 

Bo B S q 

and the variation of the functional of mechanical energy U M was presented in Section 3.3 
and rewritten as follows 



: J S : 8EdV - j 5wb*pdV - J 5wt*dS . (6.16) 

Bo Bo S a 



where the enhanced Green-Lagrange strain E = E c (u) + E (a). 
The total variation is the combination of (6.15) and (6.16) 

8U^ = fllg + 6n { * ] = 8U sttff + 5Yl ext , (6.17) 

in which 



5Il sm = J S : 5EdV - J D'SEdV , (6.18) 

Bo Bo 

SKext = ~ J 6u-b*pdV - J 5wt*dS - 1 8<t>q v dV - j 6(f>q s dS , (6.19) 



B S a B 



235 



where the second Piola-Kirchhoff stress S and the electric displacement D are determined 
by the constitutive law with the knowns E and £. 

Following the standard finite element discretization as explained in Hughes [1987] 
or Bathe [1996], we discretize the reference configuration B with numnp nodes and nel 
elements, i.e., the discretization of the reference configuration B into a collection of finite 

I 1 ne ' (e) 

element subdomains Bn , such that Bo ~ U B . 

e=l 

Besides the same interpolations used on displacements u and Green -Lagrange strains 
E°, E in Section 3.3, additional linear interpolation is used on the electric potential 0, and 
the variation and increment of electric field £ by denoting 



* iVV e) , {Mi} = -B*Sf {e) , {AS Z } = -B*A0 (e) , (6.20) 

where AT* and B 4 ' take similar forms as in the Appendix A.l. 

We apply a standard finite element procedure to the discrete weak form (6. 17) on the 
element domain £?q 

6ii^ = sn$ + su% ] = m% + = o , (6.2 n 

where the discrete weak forms of the stiffness operator 511 and the external forces 511^ 
become 

* n Kf = / * {Etf {S lj } dV + J 6 {E^Y {S^} dV - J 6{S 1 } T {^}dV, 

t$> 

(6.22) 



*n2 



= - J Su-b*pdV- J 6wt*dS- J S(j)q v dV - J Scf) q s dS , (6.23) 



B (e) 5 W B («) s (e) 



where {S lj } is the vector of the second Piola-Kirchhoff stresses in the order of 

{S ij } = [S u , S 22 , S 12 , S 33 , S 23 , S U } T , (6.24) 
and the vector of electric field {£J is as follows 

{Si} = [S U S 2 ,S 3 f , (6.25) 



236 

and the vector of electric displacement {D 1 } is given as 

{&} = [D\D 2 ,D 3 ] T . (6.26) 

To simplify the presentation, we will omit the iteration index k, in the following lineariza- 
tion procedure. 

The linearization of (6.21) with respect to the primary unknowns (d^ e \ <pt e \ a.^) is 
V (AIM) • (AS e \ A^\ Aa«) = - (SU% + 611%) , (6.27) 
where the variations SU.^ in (6.22) and 5U% in (6.23) are 

m% («*•>, ^\a^) = 5d^ T f% stlJ} + 5^ T ff stlS + fa^/M, , (6.28) 
with/S^= / B T {S*}dV, f% tiff = J B* T {D>}dV, (6.29) 

fEAs= J Q T {S tj }dV, (6.30) 

(d« 0< e >) = -UPrftyt - 6f>MU , (6.31) 
with/JL = / N T b*pdV+ J N T t*dS, (6.32) 

/SL- / N* T q v pdV + J N* T q s dS, (6.33) 

«( e ) <?(«) 

by linearizing (6.28) and (6.31), the left hand side of (6.27) becomes 

V (<m (e) ) • (AS e \ A^ e \ AqW) = 5d^ T (fcWAdW + fc#A^ + fc^AaW) 
+*0« r (fcg Ad« + fcW a^J + fcgAaW) 
+5a< e > r Ad« + fcg A0 (e) + fcgAaW) . 

(6.34) 



237 



Defining the matrices of tangent material moduli C, e, e T , e as 



C = 



Qij 



Id 



e = 



Mi 



e = 



dE kl 

dD k 
dE~j 

'dS ij 
d€ k 

dD i 



d6x6 



d 3x6 



d6x3 



d3x3 



(6.35) 
(6.36) 
(6.37) 
(6.38) 



where material moduli C, e, e T , e are expressed in the convected basis and subsequently 
arranged in matrices according to the ordering of the stress components in (6.24) and the 
electric field components in (6.25). 

From (6.29)!, (6.35) and (6.37), it follows that 

°J Mstiff 



= = I ( G " s + BT ° B ) dV - 



»(e) 



M - 



df 



(e) 

Mstiff 



d<f> 



= J B T e T B*dV , 



«<«) 



*~ - S - / BT ° gdv • 

b(«) 



(6.39) 



(6.40) 



(6.41) 



where the matrix G, the stress matrix S>, and the enhancing strain interpolation matrix Q 
are given in Section 3.4. 

From (6.30), (6.35), and (6.37), it follows that 



fcto 



(e) 



5/ 



uot 



=1* 



T CBdV 



B 



■ (e) 



df 

dcf) 



W 

.(«) 



= / g T e T B 4 'dV , 



(e) 
£45 



5/ 



(6.42) 



(6.43) 



(6.44) 



238 



From (6.29)2, (6.36), and (6.38), it follows that 



K 4>u — 



df 



to 

Estiff 



dd 



fc to 



/ B^eBdV , 

,(e) 



fc 



to 
0e* 



pto 

.(e) _ ^ firtijf 



®f ^Estiff 

dato 



k 



to 



■/ 

«<«) 



B^eQdV , 



>4> 



90 



to 



«(«> 



B* T eB*dV . 



(6.45) 



(6.46) 



(6.47) 



By substituting (6.30), (6.32), (6.33) and (6.34) into (6.27), the linearized system of 
equations on the element level is obtained for arbitrary variations of the element nodal dis- 
placement d (e) , the element nodal potential d^ e \ and the element local internal parameters 
aS^ as follows 

fc«Ad« + k^A^ + ftgAaW = , 



W 



;to 



(6.48) 
(6.49) 
(6.50) 



Vu^- 1 "- *~ t "-^q^- 1 " — y e i( J Estiff ■> ^0 

Since the enhancing strains J5 are chosen discontinuously across the element bound- 
aries, the elimination of the local internal parameters at the element level is possible. 
From (6.49), the increment of internal variable vector can be expressed as 



Aa« = - \ki e l 



(6.51) 



then substituting (6.51) to (6.48) and (6.50), the condensed system of equations in element 
(e) is as follows 



Ad^ 
A0 (e) 

where the condensed tangent stiffness matrices 



-(e) -(e) 

K uu K u<t> 

-(e) -(e) 

K <j>u K <P<p J 



.w 

M 

.to 



iu (e) _ ( k (e) _ k (e) \ k (e)]- l k (e) 
^<t>4> — \ K <H> K 4>a [ K aa\ K a4> ) i 



k ie) = 
au I > n ud> 



k 



k (e) - fc (e) \k 



(6.52) 



.to 

"a<p j 1 



(6.53) 



239 



and the condensed element residual force vectors 



r M — J ext 



J Mstiff T ^ua 



k 



r (e) - n - f (e) 4- 
' £ — Hext J Estiff ' ^(pa 



k 



e) 



-1 



EAS > 



t(e) 
EAS 



(6.54) 



After assembling the element matrices in (6.53) and element residual force vectors in 
(6.54), we obtain the incremental displacement-potential problem as follows 



K uu K 



u<j> 



Ad 



Rm 
Re 



(6.55) 

(6.56) 
(6.57) 



with K uu = Ak ( 2 , = [K^f = KkZ , K H =Ak 

e=l e=l T e=i 

nei nei 

Rm = Jkr M ] , = Arg* , 

e=l e=l 

where the action of the assembly algorithm is denoted by the assembly operator A. For 
the nonlinear dynamic response calculation, the implicit time integration schemes can be 
employed. The incremental dynamic equations are obtained by including the weak form 
of inertial forces and its linearization in (6.21) and (6.27) respectively, we refer readers to 
Section 4.2 for the details. 

6.2.2.2. Linear piezoelectric material law in convected coordinate 



The constitutive relation for linear piezoelectrics is expressed as 



S = C *. E-e T 'S , 



D = e : E + e'£ , 



(6.58) 



(6.59) 



where the stress tensor S are expressed in the Cartesian basis a* and convected basis G { as 



S = S ab a a ® a b = S l3 G 1 <8> Gj , 
we obtained the relation between components and S ab 



(6.60) 



(6.61) 



240 

where G x -Gj - Sjj, o, = a 1 . 

The strain tensor E is expressed in different basis a, and Gj as 

E = E cd a c ®a d = E kl G k ® G ; , (6.62) 

we obtained the relation between the components E cd and E kl 

E cd =(G k -a c ) {G l -a d )E kl . (6.63) 

The electric displacement vector D is expressed as 

D = D l Gi = D a a a , (6.64) 

thus the relation between the components D l and D a is 

D x = (G l -a a )b a . (6.65) 

The electric potential gradient vector £ is expressed as 

S = = £ a a a , (6.66) 

thus the relation between the components t\ and 8 a is 

£ a = (a a >G l )£ i . (6.67) 

The fourth-order elastic constitutive tensor C is as follows 

C = C abcd a a ®a b ®a c ®a d = C ijkl G t ® G^ ®G k ®G[, (6.68) 

and the component form is 

The third-order piezoelectric tensor e determined at constant strain takes the form 

e = e abc a a ® a 6 ® a c = e y *G< <g) G, ® G fc , (6.70) 



241 



and the component form is the following 



e ijk = (G l -a a ) (G j -a b ) (G k -a c ) e abc 



(6.71) 



and the second-order dielectric tensor e determined at constant strain is expressed as 



e = e ab a a ®a b = 6 l3 Gi®Gj , 



(6.72) 



thus the component form is 



e««(G i -o.) (&-a b )e ab 



(6.73) 



From (6.58) and (6.59), the relationship between the stress components and the strain com- 
ponents, the electric displacement components and the electric field components, with re- 
spect to different basis a* and G { are, respectively 



J = O Hied — e &m ) 



iabcd 



mab i 



D a = e abc E bc + e ab £ b , 



~ab i 



(6.74) 
(6.75) 



and 



gn = c ljkl E kl -e m] £ n , 



D l = e ijk E jk + e% . 



(6.76) 
(6.77) 



If we express (6.61), (6.63), (6.65), and (6.67) in the matrix form by the same com- 
ponent ordering as in (6.24) and (6.25), we obtain 



{&} - T T G {S ab } , {E cd } = T G {E kl } , 
{D 1 } = T T e {/>} , {£ a } = T e {£} , 



where the matrix T G is in Section 3.3, and the matrix T e is 



T = 



t 1 t 2 t 3 

L \ L l L \ 

A f 2 f 3 

b 2 l 2 L 2 

L ^3 ^3 ^3 . 



(6.78) 
(6.79) 



(6.80) 



242 



with the coefficients t{ = a, , G- J , i, j = 1, 2, 3. 

Since in the Cartesian coordinate, the constitutive relation of (6.74) and (6.75) in the 
vector form are expressed as follows 



{S ab } = [c abcd ] {E cd } - [e mab ] {S m } , 
{/>} = [e abc ] {E bc } + [e ab ] {£ b } , 



(6.81) 
(6.82) 



and in the convected coordinate, the constitutive relation of (6.76) and (6.77) in the vector 
form is as follows 



{E jk } + e« {Ej} , 



(6.83) 
(6.84) 



substituting (6.78) 2 and (6.79) 2 into (6.81) and (6.82), and then substituting (6.81) and 
(6.82) in (6.78)i and (6.79) r . By comparing the resulting equations with (6.83) and (6.84), 



the constitutive matrix 



Qijkl 



, and [e lj ] in the convective coordinates associated with 



basis G { are transformed from the ones in the Cartesian coordinate 



,ijk 



= T 



= T 



= T 



C 



e abc 



-ab j, 



abed 



G 



(6.85) 
(6.86) 
(6.87) 



6.2.3. Composite Solid-Shell Element 

To improve the modeling efficiency of laminated composite shells, a composite solid- 
shell element is developed here. As shown in Figure 6.1, the material layer can be stacked 
in parallel to the upper surface and lower surface of the eight-node solid-shell element. The 
element matrices are obtained by using the numerical integration, in which 2x2 Gauss 
quadrature is used in the plane of the lamina, and two Gaussian points are used for each 
material layer in the thickness direction. 



H 


c3 
1 s 

C / G 












/ B ///// 



A 



//// / 


w- ■ 











e 



x layer n 



x layer i 



layer 1 



243 



Figure 6. 1 . Composite solid-shell element: Eight-node composite solid-shell element in the 
isoparametric space, two Gaussian points (x) for each layer, and eight collocation points 
for assumed natural strain methods. 

Assuming that the thickness of layer (£) within an element remains constant 

n 

and total thickness H = of n layers is much smaller than other dimensions of the 

element, it is straightforward to find isoparametric coordinates £ 3 in the thickness direction 
of Gaussian points at each layer. For a typical layer (i), the isoparametric coordinates 
and in the thickness direction at its lower surface and upper surface are respectively 

i-l 2 » 

(6.88) 



2 1-1 2 



H: 



By using and the isoparametric coordinates of layer Gaussian points through 
the thickness direction are obtained. Accordingly, the weight factor for each integra- 
tion point at layer (i) is scaled by 



(i) h 

H 



To avoid shear-locking of the displacement formulation, we employ the assumed 
natural strain method, as applied to the four-node shell element in Dvorkin and Bathe 
[1984]. The assumed transverse shear strains are based on the the constant-linear interpo- 
lations of compatible transverse shear strains E^ 3 , E% 3 in (6.9), evaluated at the midpoints 
Q = A, B, C, D of the element boundaries with £ 3 = (Figure 6.1). 

In the case of curved thin shell structures or the nonlinear analysis, to circumvent 
the locking effect from parasitic transverse normal strain, we employ an assumed strain 
approximation for the covariant component E$ 3 of the compatible Green-Lagrangian strain 



244 



tensor, refer to Betsch and Stein [1995]. We assume bilinear interpolations of the transverse 
normal strain field, where the points Q = E,F,G,H at the corners of element midsurface 
(Figure 6.1) serve as sampling points of the compatible transverse normal strain. The 
readers are referred to Section 3.3 for more details. 

6.3. Simulation Control Design 
In this section, we present the procedure in integrating the finite element analysis 
with the control algorithm to simulate and control the response of an active structure with 
piezoelectric sensors and actuators (Figure 6.2). The static condensation is employed to 
eliminate the zero-mass degrees of freedom (dofs) associated with the electric field. A 
modal analysis is then performed to transform the coupled finite element equations of mo- 
tion into the reduced-order model in the modal coordinates. The linear quadratic regulator 
(LQR) is then employed to emulate the optimal controller by solving the Riccati equations 
from the modal state space model. 

Feedback 4> a 



Actuator 
Sensor 





Host structure 






Controller 













Sensor 4> s 



Figure 6.2. The typical active structure configuration. 
6.3.1. Finite Element System Equation of Piezoelectric Structure 

The finite element equation for the linear piezoelectric structure, without considering 
damping, is as follows 



M 




d 



+ 



K<j>u 



d 



Fext 
Qext 



(6.89) 



where stiffness matrices K uu , K u4> and are in (6.56), and external forces F ext and 
Q ext are from (6.57) without considering the internal forces, M the mass matrix. Note 
that the dofs with the electric field are "massless." 



245 



We partition the piezoelectric dofs into two parts, that is, sensor dofs and actuator 
dofs, as follows 



= 



S 
0° 



l Qext T 



s 

ext 



Q 

o a 

^ext 



(6.90) 



where the actuator voltage dofs (p a are known from the input, and the external electric flux 
Qe Xt associated with the sensor dofs, in general, is zero; the voltage dofs (f> 3 and electric 
flux Q" xt are unknowns. 

The matrices and K U(t> become 



K<j>4> — 



<t>4> 



ryss jy-aa 



i — [K U( p} T — 



jy~ss jy-aa 
4>u ** q>u 



where K S A. = 





r 




jyss 




ryaa 

u <t>. 



. and JC" = 



Accordingly, the system equations of (6.89) becomes 



MOO 





d 










1 




F ext 


?• 


► + 




TS-SS 

K u 






H 








. ii 0u 


44 


iv-aa 

K 4><t> J 









From (6.93)2, the sensor voltage increments <j> s are obtained by 



p = 



(Qext -K^d-K^) 



substituting (6.94) into (6.93) a , we obtain 

ryss 



M d + [K uu -K s ° 



ext *» 



Qext + -K"u! 



ryss 



-1 



(6.91) 
(6.92) 



(6.93) 



(6.94) 



«5 - *C </> a • (6.95) 



During the solving procedure, we calculate the displacements d by (6.95), then solve the 
sensor nodal voltage 4> s by (6.94). 



246 



There are two particular cases in (6.93), that is, only sensors existing or only actuators 
existing in the structure. If only sensors exist (i.e, <f> = <j> s ), the system equation (6.93) is 
reduced to 



M 


" 










{IV 



n uu u<t> 
jy-ss IS" s s 
■ fY 4>u ■** 4>4> 



d 

S 



Fext 
Qext 



(6.96) 



we can solve for d by setting = in (6.95), that is 

-l 



M d + K uu - Kl 



K S j?„ ) d = F ext — K. 



then solve for <p s by setting a = in (6.94), that is 



' U(j> 



jy-SS 



1 -1 



Q 



ext ' 



(6.97) 



1 (Qext -^d) 



(6.98) 



On the other hand, if there are only actuators in the structure (i.e, <f> = (f> a ), the 
system equation (6.93) becomes 



M 




d 



+ 



T/~ ry-aa 
*» 1MI ■*» u<t> 
ry-aa -ry-aa 



d 

4> a 



Fext 
Qext 



(6.99) 



we use (6.99)i to solve for d with the known <p a , that is 

M d + iir uu cZ = F ext - K™$ a , (6.100) 

then solve for the electric flux Q^ xt with (6.99) 2 , that is 

Q a ext = K a 4> a u d + K™<f> a . (6.101) 

6.3.2. Reduced-Order Model of Piezoelectric Finite Element System 

Consider a typical structure bonded with piezoelectric actuators and sensors. The 
goal of the design is to suppress unwanted vibrations and increase damping of the structure, 
which can be achieved by proper controlling of voltage signals 4> a to the actuators. 
With the external electric flux Q s ext = 0, (6.95) and (6.94) become respectively 



M d + [ K uu - K 



lie? 



ry-ss 
K 4>4> 



K'L d = F ext + 



K u<p 



ry-ss 

4><t> 



-l 



(-K£d - K™<j> a ) , 



(6.102) 
(6.103) 



247 



To compute the signal outputs s from piezoelectric sensors, we assume that there is 
no coupling between sensors and actuators such that in (6.102) and (6.103) 

K£«0, (6.104) 

and denote 

l K%) ■ (6.105) 

At the right hand side of (6.102), the actuation force vector F a , is related to the coupling 
stiffness matrix JK^J and the vector of applied voltages <p a by 

F a =-K%<p. (6.106) 

Using (6.104), (6.105), and (6.106) in (6.102) and (6.103), the governing dynamic equa- 
tions of the structure under both mechanical excitations and actuation forces can be ex- 
pressed as follows 

Md + Kd = F ext - F a , (6.107) 

and the sensor output is 

l K&d- (6.108) 

It is noted that the response of the system in (6.107) is regulated by the control voltage 
vector (f) a , which depends on the information of the states of the system measured through 
sensors. To construct a control law much more efficiently, all matrices in (6.107) should be 
transformed into diagonal forms. For the linear system, (6.107) can be decoupled by means 
of the modal transformation, which is based on the solution of the generalized eigenvalue 
problem 

X* = M*0 2 GR nxn , (6.109) 

where Vt 2 = diag [uf, •••,u$ € contains the eigenvalues, and # = [rjf v ■ • • , <0J e 
R nxn contains the eigenvectors. 



K=[K 



TfSS 

*u<t> 



K 



The cost of solving the generalized eigenvalue problem (6. 109), however, can be pro- 
hibitively high for the large size n. For structural dynamics problems, the typical modal 
analysis studies on the frequency content and spatial distribution of the excitation have 
shown that the response is controlled by a relatively small number of low frequency modes. 
On the other hand, the finite element analysis approximates the lowest frequencies and 
the associated mode shapes best, and has worse accuracy in higher frequencies and mode 
shapes. Therefore, in practice only the mode shapes with low frequencies are used for 
the dynamic response of the structure. Two iteration methods, WYD Ritz vector approach 
(Wilson, Yuan and Dickens [1982], Leger [1986]) and Lanczos approach (Leger [1986], 
Parlett [1998]), can generate a few eigenpairs in low frequencies much more effectively, 
compared to the traditional subspace iteration method (e.g., Hughes [1987, p.576]). Both 
algorithms were implemented by authors in the code of CFD-ACE+ and listed in the Ap- 
pendix B.l. For the details of the performance of the above two algorithms, please see the 
series reports by Vu-Quoc, Tan and Zhai [1998-2000]. 

The transformation is given by 

d = <&q » V r q r , d = « y r q r , (6.110) 

where * is the modal matrix and q the vector of modal coordinates for the full-order 
model (6.107), \I> r and q r contain the first n r (<C n) modes and the corresponding modal 
coordinates. The eigenvectors Vl/ r are orthogonal with respect to both the stiffness matrix 
K and the mass matrix M, and normalized to M as follows 

VjKV r = Q 2 r , *jM* r = / , (6.1 1 1) 

where 0, = diag u/jf, • • • ,uj 2 t contains the first n r eigenvalues cuf of the structure. By 
substituting (6.110) in (6.107), and premultiplying at both sides of (6.107), and then 
using (6.1 1 1), we obtained the uncoupled reduced-order model 

q T + Q 2 r q r = <i> T r F ext + f r , (6. 1 1 2) 



249 



where f T is the modal control force vector as follows 



f r =*l F a = -V 1 r K™<t> a 



(6.113) 



In general, it is not necessary to control all the first n r modes in the reduced-order 
uncoupled model of (6.112). Instead, only the first few modes are used for the control 
design. For this purpose, the n r modal equations in (6.1 12) are separated into the first n c 
equations of controlled modes and the rest n u (= n r — n c ) equations of uncontrolled modes. 
Accordingly, the modal matrix ty r , the modal coordinate vector q r , the modal control force 
vector f r , and the modal excitation force vector & r T F ext are partitioned into two parts 
as follows 



Qc 



therefore, the dynamic equation for the first n c controlled modes are 



) r ext 



* r F » 

x ext 



(6.114) 
, (6.115) 



q c + n 2 c q c = f c +* T c F ext , 



(6.116) 



where f2 c 2 = diag 
n u modes are 



, and the dynamic equation for the rest of the uncontrolled 



q u +n 2 u q u = f u +*lF ext , 



(6.117) 



where f2„ = diag 



UJ. 



n c +l> 



6.3.3. Controller Design 



The controller or control law describes the algorithm or signal processing used by 
the control process to generate the actuator signals from the sensor and command signals 
it receives. Since the 1960s, modern state-space controller design methods such as the 
linear quadratic regulator (LQR) and the linear quadratic Gaussian (LQG) (e.g., Anderson 
and Moore [1990]) have been developed in the linear time-invariant (LTI) system. In the 



250 



uncoupled system (6.1 16), the linear-quadratic regulator (LQR) is used to determine the 
modal control force for any given mode, which depends on only the modal coordinate and 
modal velocity of that mode. As a result, the independence of the open-loop equations for 
each mode is preserved for the closed-loop system. For the details, readers refer to e.g. 
Boyd and Barratt [1991]. 
Letting 



Qc 

he 



(6.118) 



2n c xl 



one can transfer the dynamic equations (6. 1 1 6) for controlled modes into state space equa- 
tions that are suitable for control design. The state space equations from (6.1 16) can be 
written in the matrix form as 



X c = AX c + Bf c + D F ext , 



(6.119) 



where 



A = 



/ 

■nl o 



B 




I 



D = 







(6.120) 



For the system described by (6.1 19), the linear feedback control law is defined as follows 



f c — - Kc X c , 



(6.121) 



where matrix K c £ K n = x2ric j s partitioned into the displacement gain matrix G d € 
R ncXnc corresponding to modal coordinates q c and the velocity gain matrix G v 6 ]R n cX«c 
corresponding to modal velocities q c 

K c =[G d G v ] , (6.122) 

substituting (6.121) into (6.1 19), it yields the closed-loop state-space equation 

X c = (A - B K c ) X c + D F ext , (6.123) 

for the response of the active control system. In terms of modal coordinates of controlled 
modes, the control force vector f c in (6.1 15) i and (6.121) takes the form 

f c = -<H c T K a u a A a = -K c X c =-(G d q c + G v q c ) . (6.124) 



251 



The optimal regulator state feedback gain matrix K c may be obtained by minimizing 
a linear-quadratic cost function J, which is defined as follows 

J = J™ (X C T QX C + f c T R f c )dt, (6. 1 25) 

where f c is the input force in (6. 1 24), Q the weight for the effectiveness, and R the weight 
for forces f c . 

The weighting matrix Q in the performance function J of (6.125) is usually assumed 
to be diagonal (Meirovitch [1990]) as follows 

Q m diag (wj, • • -,u 2 nc , 1, • • • , l) € R^ eX 2n e f (6.126) 

however, there are no general guidelines for the choice of R, the weighting matrix for 
control forces f c . A diagonal R, as assumed in many applications, may be used as 

R = diag (R u ■ ■ ■ , RrJ € M" cXnc . (6.127) 

Define the optimal gain matrix K c 

K C =R 1 B T P C , (6.128) 

which is obtained once P c is solved from the following Riccati equation 

A T P C + P C A + Q-P c BR 1 B T P c = 0, (6.129) 

where the above matrices A and B are from (6.120). 

With matrix K c in hand, the closed-loop response of the system is obtained by 
integrating (6.123). Subsequently, the control voltages cj) a for the actuators are calculated 
through (6.124) 

4> a = [V/KZ]- 1 K C X C = [tf/lCj -1 (G d q c + G v q c ) , (6.130) 

where the number of controlled modes n c and the size of vector cjf of actuators may not 
be the same, hence the inverse, ^^K^ 1 in (6.130) is usually operated by a pseudo- 
inverse process. Therefore, the modal forces f c are only approximately independent, 
depending on how close ty/K^ is to be a square matrix. 



252 



The response of the structure is calculated by combining of the contribution from the 
controlled modes * c and uncontrolled modes * u 

d a y c q c + <H u q u , (6.131) 

the sensor voltages 4> s are then approximated by using (6.108) and (6.131) 

s « - [K%] _1 K s ; u (* c g c + * u qJ , (6.132) 



and the rate of the sensor voltages <p s are obtained by differentiating (6.132) with respect 
to time 

>*-[«Sr 1 ^S(»e4. + »«M • (6.133) 

Under a given structural excitation, the structure would vibrate accordingly, and the 
distributed sensor outputs could be calculated from (6.132). Then the voltage supplied to 
actuators could be determined from the control law by using the calculated sensor outputs 
as input. Then the new state of the structure could be calculated under both the external 
excitations and the actuation voltages applied to the structure through actuators. The pa- 
rameters in (6. 127) of the controller is then modified to optimize (or tune) the performance 
for the desired closed-loop response. From the closed-loop system of (6.123), it is clear 
that the displacement feedback - G d q c modifies the stiffness, and the rate feedback of 
- G v q c modifies the damping of the open-loop system (6.120), that is 



A - B K c 



/ 

Ql — Gh — G, 



(6.134) 



The solving procedure on the above control design is given in the Appendix B.2. 

6.4. Numerical Examples 

The finite element formulations of the present low-order solid-shell element for anal- 
ysis of piezoelectric shell structure presented in previous sections have been implemented 
in the Matlab, and run on a Compaq Alpha workstation with UNIX OSF1 V5.0 910 oper- 
ating system. In each element, the mass matrix is evaluated by using the Gauss integration 



253 

2 x 2 x 2, the tangent stiffness matrix and the dynamic residual force vector are evaluated 
by using the Gauss integration 2 x 2 in the in-plane direction, and two Gauss points in the 
thickness direction for each material layer. Below we present the examples involving the 
static, dynamic analyses and active vibration control of piezoelectric shell structures. 
6.4.1. Cantilever Plate: Out-of-Plane Bending 

We present this example to show that the eight-node solid element with incompatible 
modes and the high-order twenty-node solid element (with reduced integration) suffer from 
the shear-locking in thin shell applications. 

A cantilever plate of length L = 10 and width W = 1 is subjected to the transverse 
shear loading F at the free end (Figure 3.10 and Figure 3.12). We consider various models 
with different aspect ratios L/h to compare the performances of several elements. To have 
the same level of deflection magnitude regardless of the thickness h, the applied loading 
F is set to be proportional to the thickness raised to power three (i.e., h 3 ) in the numerical 
calculation. 

The material properties are prescribed to be 

£ = 1.0xl0 7 , i/ = 0.4, (6.135) 

where E and u are the Young's modulus and the Poisson's ratio, respectively. 

A comparison of the tip deflection of the plate for different elements is shown in 
Figure 6.3. All results are normalized to the solution obtained from the geometrically- 
exact shell element (Vu-Quoc, Deng and Tan [2000]). The FE models involved are made 
of ten eight-node current solid-shell elements, ten eight-node solid elements with full in- 
tegration, ten eight-node solid elements with incompatible modes (Taylor et al. [1976]), 
five twenty-node solid elements with full integration, and five twenty-node solid elements 
with 14-point reduced integration scheme (Hoit and Krishnamurthy [1995]), all with one 
element in the thickness. It is noted that the direct use of twenty-node solid element with 
2x2x2 reduced integration scheme encounters the singularity in this problem. In the 
linear problem, the transverse loading F = 10 4 h 3 is applied at the free end. The free-tip 



254 



transverse displacement along the force direction at the corner of the midsurface of the 
plate obtained from the geometrically-exact shell element, approaches w = 3.9236 at the 
thin limit (the solution from Euler-Bernoulli beam theory w = 4). For the large deforma- 
tion case, the loading F = 5 x 10 4 /i 3 is applied in five equal load steps. The tip deflection 
in the transverse direction obtained from the geometrically-exact shell element approaches 
w — 7.41366 at the thin limit. In both linear and nonlinear situations, the present solid- 
shell element yields excellent results even for the extremely thin plate (aspect ratio = 6667 
or h = 1.5 X 10" 3 ). The eight-node solid element with full integration suffers from the se- 
vere locking. The twenty-node element (with reduced integration) and the eight-node solid 
element with incompatible modes cannot obtain correct results, especially for the large de- 
formation analysis, while the twenty-node element with 14 integration points had better 
performance than the eight-node solid element with incompatible modes and the twenty- 
node element with full integration. It is interesting to note that a twenty-node solid element 
with another 14 integration points rule by Irons [1971] gives the same unsatisfactory results 
as that with the full Gaussian integration (3x3x3 integration points), but taking almost 
half of the computational effort. 



- 5 20- noils bnck w/ 14 Rl 

- 10 8- nod* bock with Incomparibte mod* 

- 5 20-nodabnckw'FI 

- IQB-nod. brtdcw/FI 



9 0.6 



- -O -e-o 



10 geometTKatfy sua si 

-<> 10 pressol element 

520-node brkk w/ >4RI 

lOB-nodePrx 
5 20-nod* bnck w/FI 
-*- iOfl-noo»bne> 



Lib. " '° * * L/h 

Figure 6.3. Out-of-plane bending: Element performance at different aspect ratios L/h for 
both linear case (left) and geometrical nonlinear case (right). 

For flat plates undergoing small deformation, numerical results show that it is suffi- 
cient to consider the ANS treatment of only the transverse shear strain E 13 and E 23 ; the 
additional ANS treatment of the transverse normal strain E 33 does not change the numerical 
results. 



255 



6.4.2. Multilayer Composite Hyperbolical Shell 

This examples is used here to verify the correctness of the proposed composite solid- 
shell element in thin shell applications. The same example was considered in Basar et al. 
[1993] to test their shell element formulation. The shell structure consists of three layers 
with the same layer thickness y)h = h/3, which are placed symmetrically with respect to 
the middle surface. Due to symmetry, only one-eighth of the shell structure is modeled with 
FE meshes of 14 x 14 x 3 solid-shell elements (one element per material layer) and 14 x 
14 x 1 (one element through the thickness h) composite solid-shell elements respectively 
(Figure 6.4, 0° along circumferential direction). The layer material properties are E n = 
40 x 10 9 , E 22 = £ 33 = 10 9 , u l2 = 1/13 = ^23 = 0.25 , G l2 = G u = G 23 = 
0.6 x 10 9 . The analysis was carried out for two different stacking sequences: [0°/90°/0°] 
and [90°/0°/90°]. The load-displacement diagram Figure 6.5 shows that results obtained 
from the model with composite solid-shell elements (1,260 equations totally) agree with the 
refined model by having one solid-shell element for each material layer (2,520 equations 
totally). Therefore, the composite solid-shell element is accurate and more efficient to 
capture the overall global response such as the deflection for thin shells. The shell with the 
[90°/0°/90°] stacking sequence has larger deformation, and is less resistant to the loading 
than the shell with the [0°/90°/0°] stacking sequence. The computed results agree with 
those in Basar et al. [1993], where a layerwise shell element with complex rotation update 
was employed. It is noted that for the shell with the [90°/0°/90°] stacking sequence, a 
more refined mesh is needed to achieve the converged results (see also Basar et al. [1993]). 
The deformed shapes in Figure 6.6 for the final load P = 160 x 10 3 demonstrate clearly 
that large rotations and displacements are involved in this example. It is noted that for 
moderately thick shells or with nonlinear materials, the model with one element through 
thickness is incapable of accurately determining the structure response such as in-plane 
displacements, transverse shear stresses (e.g., Vu-Quoc, Tan and Mok [2002]). 



256 




Figure 6.4. Pinched multilayer composite hyperbolical shell: Undeformed mesh. 



160 



120 



100 



60 



40 



20 







— ' f- 




T 


- - v(B) 0/90/0 


9 




+ i 


- u (A) 0/90/0 


-Q 






v(B) 90/0/90 




+ * 


u(A) 90/0/90 







-* - 


v(B) 0/90/0 composite 
* u(A) 0/90/0 composite 


6 
i 




+ * 


□ v(B) 90/0/90 composite 


9 1 




+ n 
+ • 


+ u(A) 90/0/90 composite 


6 i 






i 

<p d 




+ k 




6 J 




\ \ 




i 

9 * 




+ <t 




6 I 

i 1 




+ « 




9 f 








6 i 


+ 




1 1 
9 f 


+ 




6 I 




k 

\ 


i / 
9 f 

6 i 

1 T 


+ 


• 


<Z> f 






t J 


/ 

++ ++ V » 


i i 



Displacement 



Figure 6.5. Pinched multilayer composite hyperbolical shell: Load-displacement diagrams 
from both solid-shell elements and composite solid-shell elements, v(B) is the displace- 
ment along axis Y at point B, u(A) the displacement along axis X at point A. 

6.4.3. Piezoelectric Bimorph Beam 

This numerical application is used to validate the developed piezoelectric solid-shell 

element in both an actuating and a sensing mechanism. The experiment consists of a can- 




257 



Figure 6.6. Pinched multilayer composite hyperbolical shell: Deformed shape with stack- 
ing sequence [0 o /90 o /0°] (left) and [90°/0°/90°] (right). 

tilevered piezoelectric bimorph beam with two equal polyvinylidene fluoride (PVDF) lay- 
ers bonded together, and polarized in parallel or anti-parallel directions, with the dimen- 
sions indicated in Figure 6.7. The beam is discretized into 10 equal solid-shell elements. 
The mechanical and piezoelectric properties of the PVDF are 



E = 2. x 10 9 Pa , v = 0. , 

e 3 i = e 32 = 0.0460C/m 2 , (6.136) 
Pn = V22 = P33 = 0.1062 x 10~ 9 F/m , 

where E and v are the Young's modulus and the Poisson's ratio, e 31 and e 32 are the piezo- 
electric stress coefficients, and p u ,p 2 2, and p 33 the electric permittivity coefficients, 
clamped 




Figure 6.7. Piezoelectric bimorph beam: geometry and mesh (left), and electric loading 
for anti-parallel polarization type (a) and parallel polarization type (b) (right). 



258 



For the anti-parallel polarization type (Figure 6.7(a)) and parallel polarization type (Fig- 
ure 6.7(b)), the theoretical results on the transverse deflection of the free-tip (Andersson 
and Sjogren [2001]) are calculated by 



3 e 3 i V vl 
Wa= 2EV X 



w b = 3 



(6.137) 



respectively, where h is the beam thickness, V the voltage applied on the surfaces of the 
PVDF layers. 

The present results obtained from both cases agree exactly with the theoretical re- 
sults. The deflections along the beam length for the anti-parallel polarization case are 
given in Table 6.1, in which the results obtained from the present piezoelectric solid-shell 
element are compared with that of a four-node shell element (Detwiler et al. [1995]), a 
nine-node shell element (Balamurugan and Narayanan [200 1&]), and the experiment (Ha 
et al. [1992]). 

Table 6.1. Piezoelectric bimorph beam: Deflections (xl0~ 7 m) for anti-parallel polariza- 
tion type. 



Location (m) 


4-node shell 


9-node shell 


present and theory 


experiment 


0.02 


0.139 


0.144 


0.138 




0.04 


0.547 


0.557 


0.552 




0.06 


1.135 


1.240 


1.242 




0.08 


2.198 


2.192 


2.208 




0.1 


3.416 


3.415 


3.450 


3.15 



The sensing voltage distribution of the bimorph beam with anti-parallel polarization 
under the prescribed free-tip deflection is also analyzed. The voltage distribution for a 
prescribed free-tip deflection of 0.01 (or equivalently F = 0.0254371 at the free tip) is 
given in Figure 6.8, which agrees well with the results from a laminated triangle shell 
element by Tzou and Ye [1996]. The highest sensor voltage at X = indicates that the 
largest induced-strain at the clamped end take places under the free-tip loading. 



350 



300 
250 

ST 200 
150 
100 
50 


0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

X{m) 

Figure 6.8. Piezoelectric bimorph beam: Sensor voltages along the length of bimorph 
beam. 

6.4.4. Cantilever Plate with PZT Actuators 

This problem has been studied experimentally by Crawley and Lazarus [1991], and 
used extensively for the verification of various finite element formulations (Ha et al. [1992], 
Detwiler et al. [1995] and Saravanos et al. [1997]). 

The cantilever actuator/plate system, consists of a plate with piezoelectric actuators 
symmetrically bonded to both upper and lower surfaces (Figure 6.9). For the plate of 
aluminum, a constant voltage of 252. V was applied to the outer surfaces of actuators; for 
the composite plate with lay up [0/ ± 45] s , a constant voltage of 157.6V was applied to the 
outer surfaces of actuators; and for the composite plate with lay up [+30 2 /0] s , a constant 
voltage of 188.81/ was applied. The finite element model is shown in Figure 6.10, where 
144 solid-shell elements for the host plate and 98 piezoelectric solid-shell elements for the 
actuators are used. 



259 



top surface 
midsurface 




260 

The material properties of the aluminum are 

E = 63. x 10 9 Pa, v = 0.4 , p = 2800K g/m 3 , (6.138) 

where E, v and p are the Young's modulus, the Poisson's ratio, and the mass density, 
respectively. The layer material properties for graphite-epoxy composites are prescribed to 
be 

E xl = 143 x 10 9 Pa , E 22 = E 33 = 9.7 x 10 9 Pa , 

= vis = ^23 =0.3, (6.139) 
G n = G 13 = 6. x 10 9 Pa , G 23 = 2. x 10 9 Pa , p = WOOKg/m 3 , 
and material properties for PZT (lead zirconate titanate) actuators are 
E = 63 x 10 9 Pa , v « 0.3 , 

i\\ = = 254 x 10~ 12 m/V , (6.140) 

d l5 = d 24 = 584 x l(T 12 m/V , d 33 = 374 x l(T 12 m/l/ , 

Pu = P22 = 15.3 x 10~ 9 F/m , p 33 = 15.0 x 10 _9 F/m , p = 7mOKg/m 3 , 

where E and i/ are the Young's modulus and the Poisson's ratio, d n , d 12 , d ls , d 24 , and 
d 33 are the piezoelectric strain coefficients, and p n , p 22 , and p 33 the electric permittivity 
coefficients. 

For the comparison, the dimensionless displacements are used as follows 

M 2 M a - (Mi + M 3 )/2 M 2 - M 3 

where W is the width of the plate, and M x , M 2 , and M 3 are the transverse deflections along 
axis Z at locations shown in Figure 6.9. 

The longitudinal bending w L , transverse bending w T and lateral twisting w R of alu- 
minum plate, [0/ ± 45] s and [+30 2 /0] a composite plates are shown in Figure6.1 1-6.13, 



261 




L = 292 



Figure 6.9. Cantilever plate with PZT actuator: Geometry of cantilever actuator/plate 
system (Unit:mm). 



Figure 6.10. Cantilever plate with PZT actuator: Mesh of cantilever actuator/plate system, 
respectively. In spite of the scattering of the test data, the numerical results provide ex- 
cellent accuracy as compared with the experiments. It is noted that the results from solid 
elements with incompatible mode (e.g., Ha et al. [1992]) are overstiff (locking). The re- 
sults from QUAD4 elements are too flexible (Detwiler et al. [1995]), which may be caused 
by the inability to model the 3-D structure with ply drop-offs by using single-layer shell 
elements. 

For the large deformation analysis with linear material relation, we apply the total 



262 




Figure 6.11. Cantilever plate with PZT actuator: Longitudinal bending wl (left), trans- 
verse bending w T and lateral twist w R (right) of aluminum plate. 




Figure 6.12. Cantilever plate with PZT actuator: Longitudinal bending w L (left), trans- 
verse bending w T and lateral twist w R (right) of graph ite/epoxy [0/ ± 45] s plate. 




Figure 6.13. Cantilever plate with PZT actuator: Longitudinal bending w L (left), trans- 
verse bending w T and lateral twist w R (right) of graph ite/epoxy [+30 2 /0] s plate. 

voltage 3152V in 20 equal steps. The final deformation of the structure, and the relation 
between the electric loading and the free-tip deflection M 2 are shown in Figure 6.14. Ta- 
ble 6.2, which depicts the values of the Euclidean norm of both the residual and the energy 



263 



norm in each iteration of one load step, clearly exhibits the quadratic rate of asymptotic 
convergence, which confirms the correctness of the present implementation. 




w/W 

Figure 6.14. Cantilever plate with PZT actuator: deformed shape (left) at voltage 3152K, 
and nonlinear load deflection curve (right). 

Table 6.2. Cantilever plate with PZT actuator: Convergence results for large deformation 
(residual norm, energy norm). 



Iter. 


Step 10 (V= 1576.) 


Step 20 (V=3152.) 





2.082e + 05, 2.306e + 02 


2.082£ + 05,2.320£ + 02 


1 


9.418e + 05,1.029e + 04 


6.431£ + 05,5.147£ + 03 


2 


3.937e + 03,3.636e + 00 


2.584£ + 03,3.530£ + 00 


3 


1.530e + 04,2.678e + 00 


5.616£ + 04,4.027£ + 00 


4 


9.056e + 02,1.189e + 00 


1.315£ + 02,2.025£ + 00 


5 


2.829e + 04,9.100e + 00 


7.709£ + 04, 7.789£ + 01 


6 


3.502e + 01,4.893e-01 


2.065£ + 01,5.667£-03 


7 


2.383e + 04,6.261e + 00 


1.985£ + 02,5.021£-04 


8 


3.320e + 00,3.523e-03 


6.243E - 02, 6.381£ - 08 


9 


2.052e + 02,4.557e - 04 


2.579£- 03,8.828£- 14 


10 


1.788c - 02, 2.098e - 08 


2.087£ - 06, 4.857£ - 20 


11 


1.282e- 03,1.796e- 14 




12 


1.430e-06,2.009e-20 





6.4.5. Cantilever Plate with PZT Actuator and Sensor 

Next, a thin cantilever plate with PZT actuator and sensor patches as shown in Fig- 
ure 6.15, is studied to show the efficiency of the present modeling and validity of the 
implemented control algorithm. The cantilever aluminum plate has the thickness h = 
0.965 x 10~ 3 m, the width W = 0.025m, and the length L = 0.226m (Figure 6.15). 



264 



Unit voltage is applied to the upper surface of actuator. The finite element model shown 
in Figure 6.16 uses 34 solid-shell elements for the host plate, six piezoelectric solid-shell 
elements for the actuator, and four piezoelectric solid-shell elements for the sensor. The 
material of both actuator and sensor is PZT-5H, and the thickness of actuator and sensor 
are 0.5 x 10~ 3 m and 0.25 x 10" 3 m, respectively. 




Figure 6. 15. Cantilever plate with PZT actuator and sensor: Geometry of cantilever system 
(Unit:mm). 

The material properties of the aluminum are the same as in (6.138). The material properties 
ofPZT-5H are 

E n = E 22 = 62.05 x 10 9 Pa , E 33 = 57.17 x 10 9 Pa , 

Via = 0.334 , i/i3 = "23 = 0.444 , (6.142) 

G 12 = 23.3 x 10 9 Pa , G 13 = G 23 = 23.0 x 10 9 Pa , 

e 3 i = e 32 = -6.5C/m 2 , e 33 = 23.3C/m 2 , e 15 - e 24 = 17C/m 2 , 

Pn = P22 = 0.1503 x 10- 9 F/m, p 33 = 0.13 x 10- 9 F/m, p = 7500Kg/m 3 . 
It is assumed that the PZT sensor and actuator are bonded perfectly to the plate. In Kim et 



265 



al. [1997], a complex and expensive modeling was used to calculate the static response, in 
which twenty-eight twenty-node solid elements are used to model the PZT regions includ- 
ing a part of the plate underneath the PZT patches, and twenty nine-node shell elements 
used for the remaining part of the plate structure, and two thirteen-node transition elements 
for the transition region between solid and flat-shell elements. Moreover, the element as- 
pect ratio has to be tuned for good accuracy. The comparison of the maximum deflection at 
the tip of the cantilever plate is present in Table 6.3. The present results agree well with the 
theoretical results (Hong [1992]), which were computed by applying an equivalent force 
corresponding to the actuator voltage on the structure. This example again verifies that the 
actuator performance is being simulated correctly and economically by using the present 
combination of solid-shell elements and piezoelectric solid-shell elements. 



Table 6.3. Cantilever plate with PZT actuator and sensor: Deflections (xl0~ 6 m) at free- 
tip. 





Kim et. al.[1997] 


44 present elems 


128 present elems 


theory 


w 


3.94 


3.3049 


3.4980 


3.53 


relative error (%) 


11.6 


6.64 


1.19 


0. 



For the vibration control, an step force of O.liV is applied at the free tip of the plate. 
The dynamic response of the FE model of forty-four elements is calculated by using mode 
superposition technique with the first nine eigenmodes. Among the first nine modes, the 
first four modes are controlled by LQR optimal control described in Section 6.3, by setting 
Ri = 0.01, i = 1, 4 in (6.127). The uncontrolled response and controlled dynamic response 
are shown in Figure 6.16. Both the corresponding output voltage from the sensor and input 
voltage for the actuator at certain locations along with time are presented in Figure 6.17. 

Consider the same cantilever plate in Figure 6.15 made by [0/90/ ± 45] s composite 
with the same material properties as in (6.140). Without changing anything else, we use 
thirty-four composite solid-shell elements for the plate. The uncontrolled response and 
controlled response of the free-tip are shown in Figure 6.18. Both the output voltage from 
the sensor and input voltage for the actuator at certain locations along with the time are 



266 




0.1 02 03 04 0.S 06 07 OS 

Brn» (sac) 



Figure 6.16. Cantilever plate with PIT actuator and sensor: The mesh of cantilever system 
(left), closed-loop step response and open-loop step response of aluminum plate (right). 

12 1 1 . ■ , , . . »IO* 

3 I 1 1 i 1 r ■ 




0.1 0.2 03 4 0.5 8 7 8 ' ' 1 1 ' ' ■ 

Twm (sac) 01 *■ 3 04 05 6 7 OS 

Time (MO 



Figure 6. 17. Cantilever plate with PZT actuator and sensor: Output voltage at point 2 of 
sensor (left), and input voltage at point 1 of actuator (right). 

shown in Figure 6.19. Again, the vibration of the plate is suppressed successfully. Com- 
pared with the previous aluminum plate, the vibrating magnitude of the composite plate is 
a little smaller, and the time period is shorter, which indicate the fact that the composite 
plate is stiffer and lighter than the aluminum plate. 

Finally, we want to point out in these two examples, high values of voltage are applied 
on the actuator (~ 10 4 V) to suppress the relatively large magnitude of vibration, thus 
the assumption of the linear piezoelectric material may not be appropriate and the use of 
nonlinear piezoelectric models is necessary. 



267 




Figure 6.18. Cantilever plate with PZT actuator and sensor: Closed-loop step response 
and open-loop step response of [0/90/ ± 45] s composite plate. 




Figure 6.19. Cantilever plate with PZT actuator and sensor: Output voltage at point 2 of 
sensor (left), and input voltage at point 1 of actuator (right) with [0/90/ ± 45] s composite 
plate. 



CHAPTER 7 
CLOSURE 

7.1. Conclusion 

This dissertation has addressed many computational aspects of multilayer shell struc- 
tures based on two finite element models. 

Firstly, we have developed the finite element formulation for analyzing the large de- 
formation of geometrically-exact sandwich shell model, whose governing equations were 
developed in Vu-Quoc et al. [1997]. In our formulation, the layer directors form a chain of 
rigid links connected to each other by universal joints. Finite rotations of the directors in 
every layer are allowed, with shear deformation independently accounted for in each layer. 
The thickness and the length of each layer can be arbitrary. We have derived the weak form 
of the equations of equilibrium of our sandwich shell model. The tangent stiffness matrix 
is thus obtained from the linearization of the weak form and the update of the inextensi- 
ble directors, which results in an asymptotically quadratic rate of convergence in numerical 
analysis. We have illustrated the essential features and generality of the present formulation 
by presenting several examples, including the sandwich plates with ply drop-offs. We refer 
to Vu-Quoc et al. [2001] for the dynamic computational formulation for the geometrically- 
exact sandwich shell, and to Vu-Quoc and Ebcioglu [2000*] for a generalization of the 
dynamic formulation to the multilayer case. For geometrically-exact multilayer beams 
with through-the-thickness deformation, we refer to Vu-Quoc and Ebcioglu [2000a]. 

Then, an efficient eight-node solid-shell element for the analysis of multilayer shells 
with a large range of aspect ratios and with nonlinear materials has been presented. We 
have proposed a new optimal number of EAS parameters in the formulation, together with 
the ANS method, to pass the membrane and out-of-plane bending plate patch tests and to 
remedy the volumetric locking. Furthermore, a modification for the efficient EAS proce- 



268 



269 



dure, which avoids the inverse operation at each element, was presented. We also proved 
the equivalence between various choices of the enhancing strains in tensor form, and com- 
pared them in terms of the relative efficiency. In contrast to Miehe [\99Sb], the alternative 
EAS approach using enhancing deformation gradient was reformulated in a much simpler 
manner. The nonlinear dynamic weak form and linearization have been derived based on 
the energy-momentum conserving algorithm for the current solid-shell element. Numer- 
ical damping was introduced to the time-integration algorithms for smoothing of high- 
frequency modes common in structural analysis. Although this destroyed the conservation 
properties of the algorithm, numerical simulations demonstrated that only minor deriva- 
tions resulted from small amounts of dissipation. With the enhancement on the compatible 
transverse normal strain, the full three-dimensional nonlinear constitutive models can be 
incorporated without resorting to the plane-stress assumption. Due to the parametrization 
of the displacements on the top and bottom surface of the present solid-shell element, the 
complicated finite rotation update in the stress-resultant shell models is no longer neces- 
sary. Moreover, it is convenient to model the shell contact problems and multilayer shell 
structure with geometry discontinuity such as piezoelectric patches. 

For the extension, a new eight-node piezoelectric solid-shell element for the analy- 
sis of active composite shells with a large range of aspect ratios has also been presented. 
The combination of ANS method and EAS method was used to deal with various locking 
mechanisms. The composite solid-shell element was proposed to make the 3-D analysis of 
thin composite shells even more efficient, while keeping the good accuracy. The active vi- 
bration control of multilayer plate/shell with distributed piezoelectric sensors and actuators 
was realized effectively by the linear-quadratic regulator design. 

Numerical examples confirmed that the present solid-shell element performance is 
competitive against more elaborate shell formulations. The combined use of both the EAS 
method and the ANS method in obtaining accurate results was justified. The present solid- 
shell element was proven for the following applications: 1) thick or extremely thin aspect 
ratio (=6667) in the linear and nonlinear regime; 2) isotropic material, composite lami- 



270 

nates with dissimilar material layers, and incompressible nonlinear materials; 3) the im- 
plicit/explicit dynamic analysis with/without numerical dissipation; 4) 3-D modeling of 
linear piezoelectric shell structure. 

12. Directions for Future Research 

Several directions for improvement or addition can be further investigated: 

Element technology. The development of new and improved elements is always in 
high demands. In particular, low-order triangle-type solid-shell element free of element 
deficiency (membrane locking, volumetric locking, shear locking, and thickness locking) 
and possessing good in-plane bending behavior is vital in bringing together shell analysis 
and automatic meshing techniques that rely on triangulation to fill arbitrarily shaped regions 
(Newsletter Vol.2( 1 ) of ADINA [2002]). On the other hand, for stress analysis of laminated 
composite thick plates, a new Hybrid-EAS solid element is under development (Vu-Quoc et 
al. [2002]), which can predict the interlaminar stresses accurately and satisfy the transverse 
shear stress continuity at layer interface and the vanishing transverse shear stress at free 
surfaces of the laminates. 

Constitutive models. To better characterize the actual response of structures, the 
nonlinear material models including anisotropy, hysteresis, and multi-field coupling (me- 
chanical, electrical, thermal, and magnetic) are needed to be developed and incorporated 
into the present element formulation. 

Adaptive mesh. In events involving extremely large deformation, such as metal 
forming, extrusion, and rolling, the element mesh is severely distorted so that the Jaco- 
bian determinants may become negative at quadrature points to abort the calculations. In 
addition, the conditioning of implicit analysis deteriorates and explicit stable time steps 
decrease rapidly. Therefore, the incorporation of remeshing with the present element for- 
mulation becomes necessary. 

Control and Optimization of active structure. For general nonlinear analysis of 
active shell structure, the nonlinear controller may be designed based on the current LTI 
controller (Boyd and Barratt [1991, p.45]). On the other hand, the optimal design for the 



271 



weight, size and location of piezoelectric sensors and actuators subjected to certain con- 
straints (e.g., stress failure criterion, maximum deflection) can be developed (e.g., Correiaa, 
Soares and Soares [2001], Han and Lee [1999]), based on the general-purpose sensitivity 
analysis and structural optimization theory (e.g., Giirdal, Haftka and Hajela [1999]). 



APPENDIX A 
SOLID-SHELL FORMULATION 

A.l. Finite Element Approximation of Solid-Shell Elemen t 
In the following section, we provide the detailed derivation of the finite element 
approximation in the solid-shell elements. 

The geometry in the initial configuration is 

where the position vector X = [X, Y t Z] T , the nodal position vector at upper surface 
X uI = [X uI , Y uI , Z uI ] T , the nodal position vector at lower surface X u = [X u , Y u , Zu] T . 
For eight-node solid-shell element, the two-dimensional shape functions Nj in the in-plane 
direction 

where £j and £f are the coordinates of node I. The convected basis vector Gi in the initial 

configuration is computed as d = . 

The displacement can be interpolated in the same way (A.l) 

« (o - t N > (#<*U \ [(i + e) h i (i - e) h] df = Nd^ , (a.3) 

where N — [jVj , N 2 , AT 3 , N 4 ], with Nj = Nj± [(1 + £ 3 ) 7 3 (1 - {•) J 3 ] and 7 3 being 
a3x3identity matrix,^ - [d? T , d? T , d? T , d? T ] T with d? = [d[f , dtf T } T being 
the displacements of the upper surface and of lower surface, respectively, at node I. 

The partial derivatives of the displacement field u with respect to natural coordinate 

(£\£ 2 ,aare 

272 



273 



To use a general finite element notation, the components of the second-order Green-Lagrange 
strain tensor, the second Piola-Kirchoff stress tensor and the fourth-order constitutive ten- 
sor are contained in the related matrices E, S, andC of the dimension 6 x 1 or 6 x 6, 
respectively. 

Table A. 1 . Transformation of indices from tensor to matrix form. 



Tensor Index 


11 


22 


12(21) 


33 


23(32) 


13(31) 


Matrix Index 


1 


2 


3 


4 


5 


6 



( T?c \ 



With (A.4) and the expression of the compatible Green-Lagrange strain tensor (3.15), 
can write the compatible Green-Lagrange strain E c in the vector form: 

G T X N^ + G T 2 N^ + d^ T N T e N^ 
GjN >e dU + ±d^ T N T e N^ 
G T 2 N^ + G%N,pdM + d^ T N T e N, e d^ 

{ G T lN ^ + G$N#dV + d^ T N T e N t ^ 
The strain-displacement matrix B is the derivatives of (A.5) with respect to nodal displ 



we 



E c 



F, c 



E c 

- C/ 22 



1F, C 

^12 



E c 



2£ 2 C 3 



2E C 



) = < 



(A.5) 



6x1 



ace- 



ment d (e) 



B = I 



GlN^+d^ T N T e N^ 
GjN^ + d^N^N^ 



& ) 



(A.6) 



6x24 



The derivatives of (A.6) with respect to nodal displacement d (e) are 

N T e N <e + N T e N^2 
i N T el N p + N T £3 N ci I 

V ■? * >S >s ,/ 144 X 



dS e) 



(A.7) 



24 



where the corresponding stress matrix $ should be 



$ = [S n J 24 , S 22 J 24) S 12 J 24 , S 33 J 24 , S 23 / 24 , 5 13 / 24 ] r € 



> 144x24 



(A.8) 



where J 24 is the identity matrix with the dimension 24 x 24. 

In Bischoff and Ramm [1997], they use different kinematics for the solid-shell el- 
ement, namely, the midsurface of the shell X m and the director X r which defines the 
thickness direction of the shell (see Figure 3.2). The geometry in the initial configuration 



is 



t=l v ' 



(A.9) 



where the position vector X = [X, Y, Z} T , the nodal position vector at midsurface X im = 
[Xim, Y im , Z im f, the nodal director vector X tT = [X ir , Y ir , Z ir f. For four-node solid- 
shell element, the two-dimensional shape functions jV, in the in-plane direction is the same 

ay 

as (A.2). and the convected basis G l in the initial configuration is G, = , and note that 

The displacement can be interpolated in the same way as (A.9) 

« (e 1 , e, e) = £ n (e,e) [u , eh] ^ = N & , ( a.i<» 

i=l 



275 



where the nodal displacement vector d\ e) includes the nodal displacement at the midsurface 
and the nodal difference of the director between the initial configuration and deformed 
configuration. d t (e) = \d!^ T , d^Y \ the interpolation matrix 



N = 



N\N 2 ,N\ N 4 ] , with N* = M [j 3 <e 3 / 3 ] , 



and J 3 is the identity matrix with dimension 3x3. The nodal displacement vector d (e) = 

'dl e)T ,4 e)T ,4 e)T ,4 e)T ] r 

For the Green-Lagrange strain E c , the strain-displacement matrix B and the matrix 
G, we follows the same procedure as in (A.4)-(A.8), by using the different interpolation 
matrix in (A. 10). 

In the formulation based on the deformation gradient F, the compatible convected 
basis is interpolated as 

$ " ^ = Gi + , & = = Goi + iV fl (0) & , (A.1 1) 

If we choose the five parameters for EAS method, 

ri = [4<\al?\o] T , r 2 = [4\4\o} T , r 3 - [Q t o,«<«f , ( a.i 2) 

for element (e), the EAS parameter vector is a« = [a[ e \ 4\ 4\ a<?\ a^f . Then the 
matrix form of T) in (4.79) is 



e4 } e4 e) o 
e4 ] e4 ] o 

o o e 3 4 e) 



(A. 13) 



The matrix form of T\ in (4.81) is 



3 

(<0 



<& 3 4 e) 



(A. 14) 



where indices i and j in T\ and ^ are the row index and column index, respectively. 



276 



Define the components f{ = HqC&\ where Hy are 

H n = (a\e,0, a\e, 0, o) , H 21 = (o, ofc 1 , 0, a% 2 , ) , 
H 31 = (0, 0, 0, 0, a?£ 3 ) , 12 = (a^ 1 , 0, a^ 2 , 0, o) , 
H 22 = (0, a 2 e, 0, a 2 e 2 , o) , H 32 . (o, 0, 0, 0, a 3 £ 3 ) , 
i? 13 = 0, a 2 e 2 , 0, 0) j H 23 = (o, 0, a 2 £ 2 , o) , 

H 33 = (0,0,0,0,a 3 £ 3 ) , 



and denote 



( H ^ 3x5 =9okH ki ,i,k= 1,2,3, 

where the index k uses the summation convention. 
The spatial convected basis g i is 

9i = 9i + g, , 

with g\ = G, + , g l = ff,<*M , i = 1, 2, 3, 

we define the following operators Ly, D°, Q { 



3x24 



(0,), 







The strain vector E is 



522 

&2 + #2*1 
#3*3 



#23 + #32 
I 9*13 + 931 ) 



(A. 15) 



(A. 16) 



(A.17) 
(A. 18) 

(A. 19) 
(A.20) 



(A.21) 



6x1 



277 



where E° is the same as (A.5), and = g\*gj + 9i'g] + 9i'9j- 
The strain-displacement matrix B is 

9lQi 
9IQ2 

9IQ1 + g[Q 2 



b = 



glQz 



glQz + glQ 2 
{ glQs + glQi 

and the strain-displacement matrix B is 

g[n x 
g T 2 H 2 

g\H 2 + g\H x 



gjn 3 



6x24 



6x5 



(A.22) 



(A.23) 



glH 3 + gjH 2 
{ gjH 3 + glM x J 

Similarly, the geometrical stiffness matrices G uu , G ua , G au , and G aa in (4.94) are the 
derivatives of the strain-displacement matrices B in (A.22) and B in (A.23) with respect 
to nodal displacements and internal parameters a^ e) respectively, which are 



278 



_ dB 



Qi Hi + L n 
Q 2 Hi + L22 

QjH 2 + Q T 2 H X + L l2 + L 2 i 

Qz H3 + l 33 

— T~ — T — ~ ~ 

Q2 Hz + Q 3 H 2 + L 2 z + L 32 

— T — — T — — ~ 

Qi H z + Q 3 Hi+ L 13 + L 31 



(A.25) 



144x5 



dB_ 



H^Q, + L T n 
H 2 Q 2 + L 22 

H T 2 Q l + H T X Q 2 + 1\ 2 + L T 2l 
H 3 Q 3 + L 33 

H 3 Q 2 + H 2 Q 3 + L 23 + L 32 



(A.26) 



30x24 



OB 



~T — 
H 2 if 2 

H T X H2 + H 2 H 1 
H 3 H 3 

HlH 3 + H T 3 H 2 
{ H H, + H T 3 H 1 J 



30x5 



(A.27) 



where the corresponding stress matrices $ u in (4.90) and (4.92) is the same as (A.8) and 
§ a in (4.91) and (4.93) should be 



S a - [S n l 5 , 5 22 / 5 , S l2 I 5 , 5 33 / 5 , S 23 I 5 , S 13 I, 



j 30x5 



(A.28) 



where 7 5 is the identity matrix with the dimension 5x5. 

For the simplified formulation, without the high-order term g { • g., the term 0*. in the 
strain of (A. 18) is 



9ij=9 c i '9 j +grg c j +g i 'g j , 



(A.29) 



subsequently, in (A.22), we replace Q { by Ntf in (A.23), replace 9i by g\; in (A.24), 
replace Qf Q . by (iVjlV^ + J*J5j + D?N 4 ^ ; in (A.25), replace Q t by N >v , and 
replace g { in L {j by g\; in (A.27), all terms are zero (i.e., G aa = ). 



A.2. Solution Procedure of Nonlinear Equations 
The iterative algorithm for solid-shell elements is as follows: 
1. Update on element level for iteration (k + 1) 
- nodal displacements: 



279 



(A.30) 



EAS parameters 



(ft) 



*S2] _1 ((*)*£ A (fc )dW + (fc) /g s ) . (A.31) 



2. At each gauss point of each element 
- enhancing strain 



- ANS on components of compatible strain 



^33 > ^13' -^23 ) 



element tangent stiffness 



-i 



(fc+i) 



element residual vector 



(A.32) 



(A.33) 



(A.34) 



(* + i)r (e) - (ft+D/S - (ft + i)/^ + [ ( ft + l)&2] r [( fc+ i)feS]" 1 (Hi)/JS* , (A.35) 
- save EAS arrays 



(fc+l^auj i [(fc+l)*aaj , (fc+l)J £j4 5 , (fc+l)C* 

3. Assembling from each element for (Jfe+1 jJif , {k+1) R. 

4. Solving and global convergence control 



M 



(A.36) 



(A.37) 



280 

if | {k+1) R | < Tol or | A (fc+1) u • {k+l) R \ < Tol, 
goto next time step 

else 

k = k + 1 
go to 1 
endif 

A.3. Explicit Integration Algorithm with EAS Method 

The dynamic system can be integrated over a typical time step [t n ,t n+1 ] using an 
explicit central difference scheme. 

1 . Given u n , u n , u n at time t n , 

2. Enforce essential moving b.c. on u n , u n , 

3. Update u n+1 at time t n+1 

At 2 

u n+ i = u n + u n At + u n — , (A.38) 

4. Update u n+l at time t n+1 

u n+1 = M~ 1 i^+j , (A39) 

where Rn +1 = - F™^ and EAS parameters are condensed inside each element. 

5. Update ii n+1 at time t n+1 



2 



Un+l - u n + — (u n + ?J n+1 ) , ( A 40 ) 



6. n = n + 1 and go to next time step. 

A.4. Return Mapping A lgorithm for J 2 Flow Theory with Isotropic Hardening 
The model can be integrated over a typical time step [t n , t n+l ] using a backward Euler 

difference scheme leading to the closed-form return mapping algorithm. 

Given CT\ K at t n , current C n+l , solve for C p n ~\, h n+1 , S, C at t n+1 . 

1. Trial calculation: 

N« (C^C?- 1 - A i2 /) = , with N U -C^N U = 1 , (A.41) 



281 



e l = ln -M i h* = h n , i = 1, 2,3 . (A.42) 

2. Constitutive relation: compute the elastic strain e*, principal stresses t u and moduli £ ep 
in the eigenspace form, hardening parameter h n+1 . 

3. Update for t n+1 



\ = exp(e l ) , ^^A?, (A.43) 



A 

i=l 



(A.44) 



^9iv , (A45) 

3 3 cep 



3 3 cep _ o r 



x t2 X t 2 w"Wn"®l>/- J ®iV' 3 (A.46) 
i=i t=i A i A j 



3 



+ 



£ ® W 6 (iV w ® JV y + JV« ® AT 41 ) , (A.47) 



where 



Tj/Xf ~ Tj/\ i2 



9ij ~ " \n _ jk 1 when K ? x ) ■ 



^ = 2A< 4 ~ when A * = A 5 i (A.48) 

where in the numerical calculation, the equal eigenvalues form is used when the difference 
of A* is less than a small tolerance (e.g., 10~ 7 ). 

A.5. Elastoplastic M oduli £■? 
To find out the elastoplastic moduli we can write the incremental form from 

(5.80) 

{At,} = [ef\ {Ae*} or At = £<* A e* , (A .49) 
From (5.48) and (5.60), the corresponding incremental form are 



Ae = Ae' - pAn - Apn , 



(A.50) 



282 

Ah = PAn h + A(3n h , (A.51) 

substituting (5.69)! in (A.50), then in (5.68) 1; and substituting (5.69) 2 in (A.51), then in 
(5.68) 2 , solving for At and Ay respectively, we obtain 

At = £ (A e' - APn) , Ay = -E h A(3n h , (A.52) 

where £ and £ h are defined in (5.73). 
The consistency condition requires 

Atcf) (r, y) = 0, (A.53) 

with definition of n and n h in (5.61) , the incremental consistency condition of (A.53) gives 

A4> = n • At + n h Ay = , (A.54) 

substituting (A.52)j and (A.52) 2 in (A.54), we obtain Ap in terms of A e f 

Ap^^SAe*, (A.55) 

where D is listed in (5.77). 

Then substituting (A.55) back into (A.52) 1; we have 

AT=£'*Ae t , (A.56) 
where the elastoplastic moduli £ ep is 

££P = ^ ~ D^ H ® ^ n ' (A - 57) 

A.6. Algorithmic Moduli for Return Mapping 

The constitutive relation for elastoplastic material in step 2 of A.4 is as follows: 
1. Initialize the trial status: 



e =e t ,h = h t ,P = 0, 

2. Loop on local iteration k : 



(A.58) 



283 



3. obtain derivatives of free energy ip, flow rule 



Define 



il>(e,h) , (f) fr,— J , (A.59) 



dtp dtp d<\> 36 

__ d>6 

dede ' th ~ dhdh ' * " ^ ' >/! ~ ^ • (A " 61) 



4. If ^ < to/ then 



T= aT' £ '=&&■ < A - 62 > 



exit 
Else 

5. Compute the residual 



r * =€-e* + pn, (A.63) 
r h =h-h t + Pn h , r =[r t ,r h f , (A.64) 



6. Define 



(A.65) 



£ = (£->+ pry ,4=(^ 1 +^)" 1 , 

D = n r £n + £ h n\ , (A .66) 
7. If (r T r +0 2 )* < tol then 

T = fe ' ^ = ^ ~ >3^ n ® ^ n ' (A - 67 ^ 

exit 
endif 



284 



(A.68) 



8. compute the incremental plastic parameters 

Ap = — [<f> - n T Sr t - n h B h r h ) , 

Ae = ~e- l S{r t + A0n) , (A.69) 

Ah = -8^B h ( r h + Apn h ) , (A.70) 

9. update strain and internal variables 

e = e + Ae, (A.71) 

h = h + Ah, (A.72) 

P = P + Ap. (A.73) 

endif 
ENDLoop 



APPENDIX B 
PIEZOELECTRIC SOLID-SHELL FORMULATION 

B.l. Model Reduction Algorithm 
Lanczos Method for Generalized Eigenproblem: 

Given Data : 

M n x n Mass Matrix 

K n x n Stiffness Matrix 

Triangularized Stiffness Matrix : 

K = L T D L nxn 

Choose an Arbitrary Starting Vector X : 

b= (X T M Xff 2 M - Normalization 

X i = X /b Vector one 

Solve for Additional Vectors with bi = and i = 2, ...r : 
(a) KX~i = M X j_i solve for ~Xi 
(6) a,-, =^ T M 

(c) X, = J, - a^j X (_i - fej.! X f _ 2 M - Orthog. 

(d) k = {X T i M I,) 1 ^ M - Normalization 
X i = X i/bi 

Construct Symm. Tridiagonal Matrix T (optiona l) : 
Calculate Eigenvalues and Eigenvectors of T r : 
T r Z = Z[X] 

U 2 m i/A 

Expand Eigenvectors to Full System Size : 
* = X Z 

B.2. Solving Procedure on Control Design 
The procedure for the vibration control of a linear time-invariant (LTI) system 

285 




286 



1. Solve eigenproblem (6.109) by using Lanczos method, and then form the reduced-order 
model (6.1 12), and partition the reduced system into controlled and uncontrolled parts. 

2. Form state-space system of reduced-order model with controlled modes from (6.1 16) 
and form state-space system of uncontrolled reduced-order model from (6.1 17); 

3. Obtain the optimal gain matrix K c in (6.128), where R may be adjusted for better 
performance; 

4. The closed-loop response of the reduced-order model by including the solution of 
(6.123) and (6.117); 



5. Feedback of actuator voltages in (6.130) and Sensor voltage outputs in (6.132). 



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BIOGRAPHICAL SKETCH 
Xiangguang Tan was born on 31 October 1970, in Lixian, Hunan Province, P. R. 
China. He received his Bachelor of Engineering degree in engineering mechanics from 
Huazhong University of Science and Technology in July 1992. He continued at Tsinghua 
University with graduate studies, and received his Master of Engineering degree in engi- 
neering mechanics in July 1995. Before joining the Computational Laboratory for Electro- 
magnetics and Solid Mechanics at the University of Florida, he worked as a college teacher 
at Hunan University for one year. Currently, he is pursuing the Doctor of Philosophy de- 
gree in engineering mechanics at the University of Florida. He is expecting to receive his 
Doctor of Philosophy degree in August 2002. His research interests include finite element 
method, computer software development and application, and engineering structure design 
and analysis. 



302 



I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 



Loc Vu-Quoc, Chairman 

Professor of Aerospace Engineering, 

Mechanics, and Engineering Science 



I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 

Raphael T. Haftka 

Distinguished Professor of Aerospace 
Engineering, Mechanics, and Engineering 
Science 



I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in s£ope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 




Marc Irtioij 
Professor of Civil and 



oastal Engineering 



I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 



W. Gregory Sawyer 

Assistant Professor of Mechanical Engineering 



This dissertation was submitted to the Graduate Faculty of the College of Engineering 
and to the Graduate School and was accepted as partial fulfillment of the requirements for 
the degree of Doctor of Philosophy. 

August 2002 ^/^Ww*^ l^U<^-< / ^-c^-^ 

Pramod P. Khargonekar 
Dean, College of Engineering 



Winfred M. Phillips 
Dean, Graduate School